It all depends on what kind of player you are. I have played both P and J-basses and i prefer the P-bass. It is great for classic rock and early heavy metal like Black Sabbath. The Jazz bass gets...
Owning both basses, I can tell you that the Fender Jazz does have a thinner neck than the Precision. I don't know the exact difference (measurement wise) but it's definitely a noticeable difference.