In almost every technical regard, Persona 4 does better than Persona 3. It's not perfect (minor things like the reduction of physical attacks to a single type, or the change to not being able to split up the party if you wanted [why would you want to?]), but it's far better, from a technical standpoint. The dungeons are better designed and better integrated with the story. The Social Links have more options than just romance (if their writing had been practiced enough Persona 3 would have featured this too).

However, I like Persona 3 far more. Mostly because of the scope of the story. Persona 3 is, from the start, a battle of human emotions and feelings. It's a war against the Shadows, and an exploration of their world. Persona 4 is a murder mystery, with some supernatural elements. I love the individual character dungeons, but I can't help but feel that the characters just never actually awakened to the world around them.

I love the story focus in Persona 3, because I really like exploring the huge, world-shattering things about the Persona world. In Persona 4, we only get a tiny bit of that near the end, and afterward, the characters go back to their lives. Part of why I loved Persona 4 Arena's story is because the Investigation Team finally realizes how much more is out there, and that they're a part of something much larger than just themselves or their town.