Those are three different things:
A Precision Bass has a single humbucking pickup; a single pickup bass if fine, but you lose the ability to dial-in or build a tone from combining two pickups. The Precision pickup is also a humbucking pickup, which is a big part of that solid PBass tone, which is different than . . . . .
A Jazz Bass, a two-pickup instrument, and generally, both of those are single-coil pickups. Single coils sound different than that PBass pickup, still good, and with two pickups you have a lot of possibilities tone-wise just in the mix of those two pickups, how much you turn one up or down vs. the other. A lot more possibilities than the single pickup PBass, although since they are single-coils, some noise can be an issue in some environments. And generally, the sonic signature of the Jazz pickups are different than the Precision pickup, most would think Jazz pickups are just a bit thinner, or brighter mids vs. a Precision pickup, but this is generally the case when you compare single coil and humbucking pickups.
The PJ setup is preferred by some as you combine that beefier sound of the PBass pickup with the bright sound of the JBass pickup back towards the bridge. It's the ticket for guys who like a PBass but wish they had a little more articulation in the highs and mids, or for JBass guys that wish their Jazz sounded a bit beefier. And again, you have two pickups to blend for lots of tonal variation.
There's way more to it than that (do you prefer a Jazz or Precision body shape, a Precision or Jazz neck, the many pickup choices you can make, etc.), but that's the way they line up to me.