I ask myself this question a lot and every time the answer is the same: I can't choose. I like each game for different reasons. Persona 3 was the game that not only introduced me to the SMT franchise but also cemented my love for JRPGs. Persona 4 however introduced me to Giantbomb and the wonder that is the Persona 4 Endurance Run.
Persona 4 feels more polished then Persona 3, doing things like telling you what the spells do while fusing and while outside of battle. I also feel that P4 has the better characters and better story. However there is just something inside of me that keeps telling me P3 is the better game for some reason. There isn't anything P3 did that P4 didn't do better or improve on but I still want to like P3 more the P4 and visa versa.
I don't know which one I like more. All I know is that I like them both a lot.