Everyone has a different opinion, but when it comes to quality, Persona 4 ranks superior in every single aspect and there's just no denying it.
Storyline? Yep. It's extremely well thought out, and it actually ties believably with the characters. In P3, all the way up to the last few bosses, you don't really know anything of why and what really going on.
Better characters? Much better. I've already mentioned their connection with the overall plot, but even upon individual analysis, they're far more complex and believable characters than the previous game's. The keyword here is believable.
The social links are greatly improved as well. Here, they're not just an excuse to become stronger in battle (if you think I am wrong, go to gamefaqs and get a guide on how to max every social link. Max all of P4's social links then try to do the same for P3. It will hurt).
Better art style? The art heavily remains the same, but it's slightly better crafted now. The scale of the bosses increased quite a lot. The dungeons have a ton of variety now. The environments are richer. Etc,etc.
In the end, if you haven't played too many games, the grand feel of the world about to be destroyed in P3 may have a bigger impact on you than P4's story does. But quality-wise, the better fighter here is undoubtedly P4.
I remember a phrase I read in a review of P4 not too long ago, and it pretty much sums up my thoughts about this.
Persona 4 simply does everything better. Nay, it does everything right. It's streamlined and shaped up into everything that the previous game should have been from the beginning.