Lot of different opinions for this. I've talked a lot about this on the various P5 threads but heres how I see it.
Gameplay:
P4>P5>P3
Okay yes I said P4 gameplay is better then 5, hear me out. The biggest flaw with Persona 4's gameplay is the random dungeons but if you go back and play them they are damn well near perfectly executed for what they are. None of them are very long (around 10 floors in each dungeon) so each dungeon doesn't overstay their welcome. The game gives you plenty of hints on when you are getting close to the end as well (unlike Morgana giving you very vague hints on how far you are in each Palace in P5, but we'll get to that) so its fairly easy to judge where you're at in a dungeon and how far you still have to go.
Persona 4 also doesn't have the problems P3 does like being unable to control your party members (although they fixed this in P3P) and as I said before the dungeons aren't very long unlike Tartarus which is just a boring endless slog to get through.
Persona 5 has Mementos which is basically Tartarus lite but you'd think the Palaces would make up for that but in my opinion they don't. The first two palaces are fantastic and extremely well designed as is the 4th palace but the 5th Palace on they become increasingly grandiose and pad out the time within them to an extreme degree the perfect example being the Mice sections in the 7th palace which are by far the most annoying and relentlessly boring mazes I've played in a long damn time.
The Problems with P5 don't end their however. The main way you contact enemies in Persona 5 is also pretty broken. By taking Cover enemies can't see you, even if they are literally standing right on top of you and the camera doesn't always see to let you move it around so you can target the enemies correct to get a jump on them forcing you to wait for them to move or re-position yourself, but thats the thing its one of the main mechanics and it doesn't work as well as it clearly should but even then this mechanic is completely over powered and broken and makes battles trivial because if you get the jump on enemies 99% of the time you are going to win before they can generally even attack. It makes combat trival and uninteresting because its only rarely you ever feel like your actually in danger.
The battle system itself certainly has some new features but I'd argue they aren't implemented all that well. Guns for example are an interesting inclusion but outside of having an easy way to knock down certain enemies they aren't that useful, some of the Gun skills you can unlock later are but Guns themselves never feel like important equipment. Hell I didn't ever change the default guns that any of the characters started with as the only time you really need to use them is to hit a weakness.
Going off of that we also have new elements Nuke and Psy and I do not understand the point of these elements. They already had added direct damage spells to Dark/Light instead of just Mudo/Hama making those elements more interesting but on top of that they also added Nuke and Psy neither of which are really needed and I flat out don't understand their inclusion. They work just like every other skill does they don't even have a unique mechanic to them like Dark/Light do with Hama/Mudo they are just 2 more elements because the developers wanted slightly more complexity? IT also creates an imbalance in your party because no one learns the Dark/Light elements outside of a guest character making them elements you only really use via the MC which isn't a terrible thing but it seems odd intentional to do that when previous Persona games would generally give you party members to cover each element appropriately.
I could get into this more but I think you get the idea, just because what they added is nice and new doesn't mean it holds up extremely well. P4 by comparison dropped several physical elements that P3 had while still keeping the magic system the same and it worked quite well, it never felt like you were outright unmatched at any point outside of the BS of getting a game over if the MC dies (which P5 still didn't fix!)
So yeah, I think Persona 4's gameplay is better, Persona 5 is certainly newer and has more activities and elements to its gameplay but I don't think all of those changes are for the better and personally I'd rather go through all of fucking Tartarus again instead of doing Palace 7 again, and neither are those experiences that I enjoyed.
Story:
P4>P3>P5
Persona 4's story centered around a murder mystery is immensely well told and interesting and even after the culprit is revealed (no matter when you figured it out, it couldn't make sense until you spoke to Namatame and he filled in the gaps of things you didn't know) the game still nails its execution to the normal ending. Then things get really interesting as during your final day you have the option of heading down the true ending path and damn does it get interesting. The ultimate reveal of Izanami and the rumble of the controller at the beginning of the game (Ps2 only sadly, sorry Vita) coming back during the very last day of the game is so well done. Even IF you thought something was strange back at the beginning by this point you are likely well over 100 hours of playtime and no fucking way you still remember that but when she does it again HOLY SHIT DID IT ALL MAKE SENSE! That is how you do fucking foreshadowing from the beginning! The final arc of the game is about understanding the truth and the single most badass moment in the series is Yu taking off his glasses (symbolizing he can see the truth even through the fog caused by Izanami) and then summoning Izanagi-no-Okami to finish her off.
The twists in this game are all well done and foreshadowed extremely well what with what I already mentioned plus the story of Izanagi and Izanami being told to you during the school trip. It also ties in the cast of the game extremely well with whats going on (and I'll get more into them later) and its all just executed perfectly.
Persona 3's story is also good but more predictable then I expected. I do want to replay through it again, it certainly has the most interesting ending but I felt the pace was off in places and the villains just weren't that interesting, the evil team of Persona users was just awful outside of the development it gives Junpei and Chidori.
Persona 5's story is... a mess. It starts out extremely strong in the first arc but each new arc sets up the same pattern that doesn't change up until the very end. Then once you get to the final arc of the game it largely repeats P4 with several twists that just aren't done as well or foreshadowed properly (as much as I love the Igor twist its really not foreshadowed well at all unless you know the voice actor stuff) and what could have a super interesting twist villain ends up going out in the lamest way possible and isn't even a hard boss fight or anything, it feels like a completely wasted character in the end and brings down the previous twist. The true villain of the game is just not very interesting and echo's Izanami and Nyx before it without ever being foreshadowed or having any kind of real build up to it. It's all done in a very grandiose way which visually is cool but doesn't escape from how poorly this whole narrative played out. The first part of the ending felt extremely forced (why do you need to turn yourself in if Shido had a change of heart and confessed??) and while touching it didn't feel like the game really earned it at that point. The second part of the ending is literally Persona 4's just replace train ride with car ride.
Characters:
P4>P5>P3
I've talked at length about how much I love Persona 4's characters so as this is already going long I'll make this single point to solidify my decision here. The characters of Persona 4 feel like real friends that know and truly care about each other. The way the spin off games talk about how the bonds they created will never be broken feels very accurate because of how well Persona 4 does at developing the cast as a group of friends. You see them develop as a group on multiple events both in the original and in the Golden remake with the Camping Trip, the adventures of Mystery Food X, the school trip where half the cast gets drunk (on the atmosphere XD), the school festival and cross dressing beauty pageant, ect. These are a real group of friends that not only work together but also really love each other and each of them is better off for knowing each other.
The Persona 5 and Persona 3 casts just don't have that. Both casts feel more like colleagues who work together but you never really get a single that they truly care about each other. Some times the games will force an emotional moment on you but as a group they just aren't well developed. Both games have group events in P4 does but they are generally lacking in group development or don't end up using the cast the way they should. Persona 5's events are used more to develop the plot then they are to actually have character development and it feels like a complete waste doing what they ultimately did with them instead of developing the group as a whole. Also P5 introduces Haru way to late into the game, it would be one thing if she was introduced earlier on and you could have built up her S-link ahead of time and then brought her into the fold but as it was done you get like 1 or 2 group scenes with her in it before the ending? One of those ends up being used on the plot like I mentioned before so it just feels extremely wasteful as I really liked her character.
Music:
P4>P5>P3
This is honestly just personal taste more then anything, I just didn't like P3's music that much and while P5 has some great music some of it like the Mementos music is burned into my brain in the worst possible way it could be. P4 also has a music/dancing spin off game largely because of the excellent music so theirs that.
Overall:
P4>P5>P3
Persona 3 getting a proper remake and getting the golden treatment might make it better then P5 honestly but as it stands this is how I see it. I don't think Persona 4 is really lacking in anything, I can see why people say the gameplay of P5 is better but after replaying through P4G again I just can't agree. It might be flashier and do away with the randomized dungeons but I don't think that alone makes it better, the Palaces in P5 start off absolutely fantastic but the later Palaces drag on and on and take hours to complete largely do to excessive padding, that isn't fun and I'll take a 10 floor random dungeon over those any day. Even if you see the gameplay in P5 being better the story and characters of Persona 4 are much more interesting and well developed and personally I just hated how many things P5 just reused from Persona 4 by the end of the game with its story elements.
Persona 4 is still the best in the series and while P5 is also good if not great its still lacking and feels rushed. The story has pacing problems among other things and I just couldn't get attached to the characters in the same way largely because they never develop as a group very much and there is clear hostility among the group throughout most of the game that just wasn't really needed and didn't help to develop the cast very well.