Ahh. Well the only major thing that Persona has that SMT doesn't is the social link stuff but even that's not present here. As for how battles are in Persona compared to SMT: along with how you get press turns when you hit an enemy weakness, Persona adds in a mechanic that includes knocking down an enemy and you can attack it again to make it dizzy (basically rendering the enemy unable to do anything for its next turn). If you can knock down all enemies during your turn, you get the opportunity to do an All-Out Attack (added in P3) which just does extra major damage. But I'll ease you into how that works later.
EDIT: Oh, and if you don't know about the Arcana, they're basically demon races for Persona.