The main battle system in terms of combat is called the "1 More!" system, and it's slightly different in each.
The core idea of it is your characters and enemies are strong to some elemental attacks, and weak to others. If you hit a weakness, you do more damage, knock the opponent down, and are able to immediately attack again with the same character. The other added benefit is by knocking down all of the enemies, you can initiate an "All Out Attack" which is all of your characters dogpiling on the all of the enemies, doing a ton of damage to each one(in a game where mana conservation is somewhat important, the ability to do free area-of-effect damage is important).
In Persona 3, something knocked down had to use up a turn to get back on their feet again, so it was effectively wasting a turn. Also, to get the benefit of "1 More!", you have to ONLY hit an enemy's weakness. In other words, if 3 are weak to fire, and one is weak to wind, using a Maragi(all fire) spell will hit 3 and knock them down, but not knock down the 4th, so no free turn(this also happens if a spell misses).
In Persona 4, being knocked down doesn't waste a turn(instead attacking something on the ground already has a chance to "Dizzy" it, which is the turn waster), and you only need to touch an enemy with a weakness to go again.
Of course, in Persona 3:FES you have no direct control of your party, only telling them what you'd like them to focus on(knocking enemies down, healing/supporting, going all out, conserving SP, etc), while you could control every action of every party member of 4.
There's more streamlining with 4 other than this, like every party member in 4 being a social link which wasn't the case in 3, but the combat is a major hurdle for some.