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. As far as live playing, you will notice a difference depending on the size of the room that you are playing in. Also, style of music that you will be playing can be a determining factor. A jazz bass will give you more of a modern tone, is better for suited for funk-style playing such as slapping, etc. 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.