Well a jazz bass will always give you versatility. For hip hop many bassists use jazzes, you can hear alot of live jazz bass on some of lil jon's cuts (a 5 string Sadowsky played by LaMarquis Jefferson, hes playing midi bass on lil jon's latest album though i think) and also the bass player on All Eyez On Me by 2pac played a jazz bass i believe. It can go from smooth to agressive pretty easily. The slap sound is also very clean and snappy which would definately come in handy at times in r&b and hip hop too.

A p-bass is not as versatile, but the one sound it makes has a way of just sitting right in the mix. Some producers dont care for the heavy, raw old school sound and feel that a jazz bass is easier to work with, but like all basses it really depends on your technique and style. Some of Jill Scott's tracks have p basses on them. "Whatever" is the best example i can think of for her. Another fan of the pbass is D'angelo its on quite a few his songs from Voodoo. Chicken Grease and just about all the other old school sounding bass sounds are a p-bass. Pino Palladino is the player D'angelos voodoo album and Whatever by Jill scott. He uses flatwounds on his p-bass.

a 3rd option would be a p/j bass with a precision pickup at the neck and a jazz pickup at the bridge. This will give you the most versatility. Its easy to get a deep sound that is still clear. Check out Stankonia by Outkast. lots of tracks with a p/j including Ms. Jackson and So Fresh So Clean, both played by a dude using a p/j (Preston Crump)

So it really comes down to what bass will emphasize the particular style you are going for, but you cant go wrong with either. The tracks i mentioned above should help you understand some of the capabilities of each of thhose basses.

This is good advice too, go to your music store and try both. The feel of a bass can be more of a deciding factor than the sound. Jazz basses have slimmer necks and pbasses feel chunkier.

You can eq the hell out of it but what really makes those basses sound like that is where the pickups are located, the type of pickups they use, and the type of wood too. Thats something you wont really be able to influence, but you can do some crazy things to the tone of a bass. Just dont expect to always be relying on eq to give you the tone you want or make one bass sound like another.

If the bass is just for studio i would reccomend a passive bass with just volume and pickup tone/blending controls. An active bass with an eq on it tends to give a much hotter signal and if you have not dealt with mixing live bass alot it can be a pain. Also the pickups tend to be much more sensitive to stuff like string squeaking and your fingers hitting the pickup, so unless your technique is on point you may find yourself doing alot of takes. On a passive bass string noise isnt as noticeable and it can even enhance the "live" feel of a bass track. These are some of the reasons that hip hop producers prefer passive basses and that alot of their samples are simple, old school ones. Some producers do use active basses though, Lil Jon's bassist uses an active 5 string jazz bass and Dr. Dre has had active basses on some cuts. They can be useful when you want the bass to really dominate the mix and influence the direction of the beat.