I've played both in jazz bands, and I can get a good tone out of either. For swing and more straight ahead stuff where I'm mainly playing 4 on the floor with the kick drum, I preferred the thicker tone that the Precision gave me. I can come close soloing the neck pickup of a jazz, but the same low mid character just isn't there.
When it came to playing more fusion stuff, I liked the growl of a jazz bass, but I could also pass using a P-bass any day of the week. It's more about you technique, and not to mention eqing, that will determine your sound.
A Precision is a very versatile instrument when it comes to tone. Just by moving where I pluck, and possibly plucking with my thumb, I can fatten up a tone, soften the attack on a growly tone, and basically achieve any degree of "cut" or what have you that I wanted.
Get to know your bass. Chock it up to the old saying "Fear the man who only owns one gun." I guarantee that a Precision can slug it out just as well as any other "jazz bass."