Well, of course the US Precision Bass is _the_ bass to which all others are compared. How could one go wrong with an American Precision?
The Jazz bass plays differently due to a somewhat skinnier neck and it has a broader range of sounds due to having two pickups, which are placed at different spots on the string track. It seems to me to be more of a shredder's bass than the Precision.
I've never owned a Jazz Bass, but have had a few Precisions. They suit my idea of what a bass should sound like, which may be more traditional than is current. I just ordered some .105" flatwounds for my near-mint 1970 Precision this morning, which may tell you a lot about what I want to hear out of a bass.
An interesting compromise is the American Series "Hotrodded" Precision Bass
http://www.fender.com/electricbasses/pbass/hotrodamerseries/ashotrodpbass.html
...which has a Jazz Bass pickup placed where it would be on the bridge position on a Jazz Bass. This bass is a total killer. Though normally very pricey at nearly $1600 list, the US Guitar Center chain had them on a one-day-only super-sale for $499 at the beginning of September. I bagged one _exactly_ like the one in the picture (right down to the grain pattern) and I'm amazed.
The graphite-reinforced neck is remarkably rigid, giving great sustain and no weird resonances or dead spots, as well as both Precision and Jazz Bass sounds from the pickups.