Basically, beyond its emphasis on story, story mode tries to make everything as un-Etrian as possible. You're saddled with the same five characters for the entire game, and unfortunately, even though you can technically reclass them, they're stuck with their original stats (so a physical attacker with a magic class would basically be useless).
Speaking of classes, story mode is the only way to get the highlander and gunner classes. Honestly, though, they're not that amazing, even though the gunner gets the utterly broken action boost ability (which allows you to use the same attack three times on the next turn with no penalty). There's also an exclusive dungeon you revisit multiple times, but it's so short and uneventful that you won't be missing much.
Everything else is basically the same, but classic has the benefit of rebalancing the original classes and fixing old glitches. I really didn't care for the extra (and pretty generic) story at all, so I'd much rather have the freedom to make a party out of characters that suit my playstyle. It also helps that you no longer need to unlock the hexer or ronin classes, so you have even more freedom in what you can do.
Plus, based on what I've read off Japanese wikis, Persona Q basically does what Untold 1's story mode tried to do for the series, but much better.