Persona 4 definitely has the better game mechanics and story mechanics. It simply plays better overall.

4 is better, but you are better off finishing persona 3 and then 4. That way you can get the significance of a certain event on 4.
Also, this way you get to enjoy both these great games. :)

I played 4 > 3 (both portable versions) and I have to say 4 is definitely the better game and the best way to get introduced in the series. You might miss out in some references during the 4th game, but I feel the impact will be the same if you play the third game afterwards. I was like "Oh damn! I remember that event/person from Persona 4!". So the cameos/references work both ways, which is awesome.

ALSO, if you want an non-frustrating experience Persona 3 Portable is arguably the better game because you can control your allies in that one (while in FES you can't). It's also much more straight to the point.