[I haven't played FF1-3]
FFIV: This game honestly reminds me a great deal of FFXIII. There's much more "world" to it, but with regard to battle design, it's very much about constantly switching up your party on you and forcing you to find a new rhythm. The battles are also very much about attrition (managing a good balance of HP/MP loss, buffs/debuffs, and damage output) and pretty much every time the game changes your party up on you, you need to radically shift your strategies. This turns each boss battle into something of a puzzle based on finding the correct balance of these factors (and, often, attacking a series of targets in the correct order), and it's also very much designed around the fact that Cecil is a Paladin (he's there for damage output AND party defense AND healing) and therefore as flexible as he needs to be for each party configuration in the game. FFIV does a better job than FFXIII of making sure that this feels like a part of the main game and not like tutorial; in many ways FFIV doesn't *fully* open up until you get your final party into place, but it does a great job of disguising that along the way. However, FFIV probably has the dullest ability/character progression system in the series. It *does* do possibly the best job in the series of making each new weapon you obtain feel like it makes a genuine difference, though - each new piece of equipment feels like a major, important upgrade. The debut of the ATB system is also quite important. But the way that each different party configuration makes a serious and real impact on how you handle parties is a big big deal.
FFIV DS does a great job of fixing the character progression system with the Augment system - which adds a LOT of dimension to the game. For the most part it's still very much attrition-based, and boss battles are still about attrition management and attacking a series of targets in the correct order. FFIV DS is *very, very heavily* oriented toward people who have already played FFIV - many boss battles are designed specifically to fuck you up extra hard if you use the old familiar strategy from the original game - and it expects its player to be a veteran of JRPGs, which is why it's one of the toughest FF games released in a long time. The nature of the augment system also makes it especially opaque to people not using a walkthrough to know how they ought to be building their party. It's a fantastic game as fanservice, but it continues to share most of the flaws of the original and is less accessible to boot.
FFV marks a major leap forward for the series in terms of progression. The Job System in FFV (marked by its much greater sense of customization than FFIII's due to the way you can mix and match abilities) is a brilliant one. However, as a result of the new version of the Job system, it's the first game in the series to introduce a heavy concept of 'breakability,' one that is both a lot of fun and tends to make games way, way too easy once you figure out some secret combination of abilities and equipment. However, the game is extraordinarily replayable and gives you tons of options for solving your problems. If FFIV's sense of progress was defined by getting new party members and finding new weapons, FFV's is all about getting (and mastering) new jobs and leveling to specific new abilities. Especially because equipment is very limited to job type, the class being equipped tends to overshadow the importance of new equipment in the game. FFV is all about dishing out physical damage, though - mages are fantastic for crushing certain bosses, but the sheer utility of dual-wielding or the Monk class can't be beat. It's also one of the first games in the series to really let you cripple the shit out of your enemies with status attacks, which is a big part of what makes it so flexible and replayable. This one's a total dud as far as the story is concerned, and equipment that isn't a Protect Ring or Ribbon never feels particularly important, but it does a great job of making each class feel genuinely useful. FFV takes the flexibility that Cecil has in FFIV and extends it to the entire party. This results in each character's "personality" not really making its way through to battle, and when FFV goes through a major death and the switch from Galuf to his horrid granddaughter Krile, your party feels precisely no different because the characters are so interchangeable.
FFVI goes back toward FFIV in terms of making each character feel more unique - however, it's very much like FFV too, in the sense that they're all capable of massive damage output, great defense, and plenty of general healing and support after a certain point in the game. FFVI's character differences are generally cosmetic - input methods aside, Setzer's "Royal Flush" attack and Edgar's "Flash" attack are barely different in function - but they make all the difference in terms of giving everyone who plays it their favorite set of party members. Regardless of whether the characters all truly *function* differently, they all *feel* different. FFVI is the one game in the series that really goes the full distance in terms of having its characters express their personalities in battle; the balance of the game suffers for it and mechanically they're not all that well-differentiated at all, but each one FEELS different, and that's why the game is near and dear to the hearts of so many. Its soundtrack also leans much more heavily on the device of leitmotif than most, and it's quite telling that FFVI's end credits music is mostly composed of a nostalgic final visit to every single character's musical theme. FFVI is much, much more about characters than it is about story. This is especially obvious in the second half of the game. The game's plot frankly falls apart at the end of the World of Balance; in the World of Ruin, the game is all about hunting down your party members, completing fun sidequests (many of which are about resolving problems for your party members), and finally going after Kefka. But there are some pretty big flaws in the World of Ruin: because the game can't predict which characters you'll have with you during certain cutscenes/dialogue segments, it just takes certain lines and shoves them in the mouth of whichever characters happen to be present. This has certain obviously weird consequences (there are a few spots in the World of Ruin where you can have Gau, Gogo, and/or Umaro lucidly speaking lines that clearly shouldn't be emerging from their ordinarily mute or inarticulate mouths), but beyond the especially silly examples, it means that character development, as much as it's the focus of the second game, is severely hampered for the characters that aren't the actual focus of a given sidequest. (This is an issue that Square-Enix sort of tried to revisit in Chrono Cross, with a sort of syntax modifier that altered the sound of certain lines depending on who was speaking them, to decidedly mixed results). FFVI takes some of the modifiers that existed in the Job System in FFV and instead incorporates them into the Relic/accessory system, allowing for a good deal of customization; the Esper system (accessible maybe a third of the way into the game) allows you to teach magic to the entire party. FFVI is very much about dealing magic damage most of the time until you get to some of the serious multi-hit physical attack stuff later in the game. Relics and accessories have the effect of making characters mostly interchangeable, but the amounts of difference that persist keep them feeling quite different from each other even though the difficulty takes a nosedive when the Espers are introduced and disappears altogether once you hit the World of Ruin.
FFVII makes the grand leap into 3D for the series. In many ways it does a lot less than FFVI to differentiate between characters, but in some ways it does a great deal more. It goes much farther than FFIV or FFVI does in terms of keeping each character's equipment set different from the rest (the characters in the game share absolutely zero weapons with each other, a first for the series), and is the first game in the series to introduce truly unique "ultimate" weapons for each character that aren't just stronger than their other weapons but are actually mechanically differentiated in some special way (the best-known example using mechanics similar to the Atma Weapon and Valiant Knife from FFVI, which deal damage that is heavily dependent on the user's HP), and easily the most simplified equipment system in any of the FF games up until this point. FFVII does away with the always-accessible character-specific attacks from FFVI, but opts instead for much flashier and more noticeable/memorable per-character "Limit Breaks" - an evolved version of FFVI's rarely-seen Desperation Attacks. These are generally just things that serve the function of pumping out a certain amount of damage (though sometimes in different ways - Vincent's limit breaks are somewhat more akin to the specialized-berserker forms seen in Mog and Gau in FFVI). The Materia system's a very interesting way of making sure each character's stats are roughly balanced out to their suitabilities for battle and it offers incredible depths of flexibility, though it suffers from what was at this point becoming a common problem in the series: that of effectively turning each character into a clone of the others. But every time you get a new materia in the game, it feels like a big damn deal and makes a noticeable difference in how powerful your party is, which is important. This raises the most important point of FFVII --- Aeris is the most heavily-differentiated character in the game. She's the only character who straight-up sucks at physical attacks and vastly overpowers everyone else at magic use, she's the only character (barring one single Limit Break of Yuffie's) with a set of defensive limit breaks. And the game kills her and never brings her back or replaces her with someone who plays a similar role. It's a fascinating design choice, and her absence is felt all the way through the end of the game.
FFVIII makes a lot of very interesting design choices and is one of the most experimental games in the series. Its bizarre leveling system makes the difficulty of the game go up precisely in proportion to how much leveling you've done; it's easily the most breakable FF game, albeit in strongly counterintuitive ways; it even makes choices like having a main character who's effectively immune to Blind status and is incapable of dealing critical 2x damage, but who is capable of dealing 1.5x damage with every single hit for the whole game. It's also easily got the simplest equipment system in the whole series (outside of unlocking Squall's limit breaks and unlocking Blind-immunity/perfect accuracy for Selphie, your weapons have only an extremely minor effect on your power, and they're often pretty complicated to obtain). FFVIII's ability/growth systems are bizarre and radically experimental, and in many ways more geared toward minmaxing powergamers than toward traditional JRPG gameplay. It also strikes a fascinating balance between FFVI's Desperation Attacks and FFVII's Limit Breaks, giving overpowered and cinematic abilities, but allowing for a very interesting risk/reward system that lets you give yourself near-constant access to your powerful Limit Break abilities in exchange for making your party vastly more vulnerable (though this system is seriously breakable with the Aura spell, which is a damn shame). FFVIII's one of the most-flawed games in the series but it's a fascinating entry. It's also the first FF game to introduce a minigame that runs throughout the entire game: Triple Triad. Triple Triad is great fun *and* it's a way to cheese a huge amount of the game by giving yourself access to extraordinarily powerful spells, abilities, and items earlier than you'd be able to obtain them otherwise. Again, FFVIII is a game made from the ground up for people who look at game systems as something to break and turn to their own advantage.
FFIX is the game where the venerable ATB system really started to show its age, and arguably the primary impetus for the more radical speed-focused alterations to the battle system that began with FFX. FFIX makes the mistake of NEVER pausing the ATB timer (FFIV and FFV pause it anytime any move is being executed; FFVI-FFVIII pause it during more elaborate/cinematic attacks), which results in turns getting queued up minutes in advance and often in reaction to situations that are no longer applicable. It's also the first mainline FF game since FFVI to have five pieces of equipment per character and to allow some characters to share weapon types. Its equipment/ability system makes obtaining equipment a huge deal, and actually allows for a surprising amount of character flexibility (in terms of being able to equip and use any abilities on your equipped items regardless of whether you've yet permanently learned them from the piece of equipment). In my own playthroughs it often trends toward completionist grinding in order to prioritize learning all abilities, I must confess. Its assorted party members feel like a set of functions rather than a set of personalities, however. It's also got heavily differentiated roles for party members in a way that's much more akin to FFIV than FFVI (though it also introduces the concept of the white mage/summoner, a combination that would pop right back up in FFX and guarantees that every single party member is capable of being a damage dealer). Another thing it has in common with Final Fantasy X is the idea of the lead character as a fast/light attack character who gets tons of turns. FFIX's Ability Stones passive-ability system is pretty primitive and it can often feel difficult to really optimize your equipment in the game, and its stat-growth system is opaque as opaque can get; but it's a great effort at a classic FF that manages to let you go on the full offense while also making your party makeup truly matter instead of merely being an aesthetic preference (though, like FFVI, this aspect falls apart more and more the further you get into the game and everybody becomes an offensive powerhouse with Auto-Regen rendering them practically unkillable). The Trance system's an interesting one, but mostly ends up being a distraction as your characters
FFX is the last FF game from before the Square/Enix merger and it's arguably the first "modern" FF in many ways (though some would put FFVI or FFVII in that position - go figure that they're the three most commonly cited as people's first/most impactful FF game). It adds (dodgy) voice acting, a fully 3D world (albeit one without camera control), and removes the world map in favor of a more strictly (and more honestly?) linear journey (albeit one well-justified by the story). The Sphere Grid and an equipment system that isn't based around weapons having stats of their own is a pretty interesting departure from previous games, but one I find really interesting. Every +4 you get to a major stat in FFX *really* makes itself felt - the Sphere Grid does a great job of intimately acquainting the player with each character's strengths and weaknesses. Final Fantasy X does a curious job of specializing each character VERY heavily for the bulk of the main game (and, unfortunately, giving too many palette-swapped fodder enemies for one-hit kills - there are plenty of enemies that don't fit that mold in the game but there are enough of them that that's what people remember most, and FFX is decidedly not a very challenging game) while making them EXTREMELY identical in the endgame (not even allowing for customized loadouts like FFVII or FFVIII - everybody can and will learn every single non-Overdrive non-Summon ability). It's also the first FF game to do away with in-battle "rows," which have never returned to the series since. But arguably the biggest and most important difference FFX introduced to the series was the CTB - not because future games would use it, but because it heralded the end of the ATB's streak of use in FF games. The CTB feels a great deal like the turn-order menu in Final Fantasy Tactics, and is a tick-based system that lets you preview the upcoming turn order, manipulate that turn order, and act with some foreknowledge of what's happening next. This has the effect of speeding up the game somewhat (no waiting for bars to fill up, immediate feedback) while also giving you time to strategize. Consequently, FFX is *about* immediate responses to the situation and strategizing, and it's very much built around single, decisive turns. Hence the hard-counter enemies that get one-hit-killed by the "correct" character; hence the Overkill system (which is all about delivering tons of damage in one single turn); hence a system of making more powerful moves generally delay a character's next turn. FFX is well-realized in a number of great ways, which is what makes it a bummer that its ultimate weapons are generally unlocked/powered up in some pretty silly and unfun ways (which sparked a lot of complaint when the game came out and again when it was rereleased recently), and a bummer that the fantastic and versatile swiss-army-knife party you have access to over the course of the game (which are well-realized in battle with individual lines they speak during battle depending on context and movesets and statistical tendencies that stay well-differentiated for the course of the story) turn into a bunch of clones once you max them all out, drastically discouraging any need to switch them in and out during battle. One underrated thing that FFX did extraordinarily well, I ought to mention, is that nearly every single story-based boss battle in it is actually a part of the plot instead of just being a strong monster that shows up at a particular point. They're mostly Sinspawn, Seymour, or Yevon-related; the only real exceptions are the early enemies Klikk, Geosgaeno, and whatever the octopus is called, and then the Spherimorph in Macalania Woods. Every other boss fight is directly tied to the story and manages to help Spira feel like the most well-told and cohesive FF world we've ever seen.
FFXII is in many ways a polar opposite of FFX. Boasting some of the best voice acting and localization we've seen in video games, full camera control, and oodles of what could charitably be called freedom (or uncharitably called a lack of direction), it's a gorgeous, extraordinarily ambitious, sloppy, sprawling counterpoint to FFX's honed, toned-down focus.
