The jazz has a one piece bar shaped pickup nearer the neck.
P bass has a brighter, crisper sound.
Played on more recordings than any other instrument in the world, I'll bet. Fender and MANY others make a model with P and J style pickups the best of both worlds. There are several subtle differences between the two. Actually, its the exact opposite. For recording, the P Bass is the standard of the industry due to the simple electronics design that allows the player to turn the volume full without any noise from the pickups.