I never played Persona 4, so some of the fanservice appeal of this game is probably lost on me. The only Persona 3 characters are Mitsuru, Akihiko, Aigis, and Elizabeth. I wouldn't be surprised if Atlus released more characters via DLC or in an updated version of the game.
The story mode is done as a visual novel. It's time-consuming but not actually very comprehensive, and it's largely non-interactive. Most of your time "playing" is really spent reading your selected character's text narration atop static backgrounds and character portraits. Seems pretty mind-numbing to me. The story itself is supposed to be a sequel to Persona 4 (which, again, doesn't mean much to me, since I never played that game), but, from what I've seen of it, it seems to be a typical fighting game story, as characters just run into and start fighting each other for fairly contrived reasons.
The fighting engine is a Guilty Gear derivative, also in the same vein as other "anime fighters," such as BlazBlue, Melty Blood, and Arcana Heart. Not sure how this came to be its own distinctly identifiable sub-genre, but it's much faster and looser than Street Fighter or King of Fighters, and it's got the usual air-dashes, double-jumps, and all manner of systems and sub-systems. Frankly, fighting games these days are just getting too convoluted for me.