When I first bought a bass I went for a Jazz because I read on various forum posts people saying that the Jazz can be made to pretty much sound like a P but a P can't be made to sound like a Jazz.
Now that I've had both for a while I know that it's absolutely not true that the J can sound like a P - it always sounds completely different regardless of settings.
So, whichever one you prefer (if you can't have both), I just feel it will be helpful to anyone coming across this thread in future to know not to go for a J expecting it to be the best of both worlds.