I own a Jazz bass and also a P/J bass. First don't compare the P/J bass to a Jazz bass. They both sounded different in my ears but I choose the P/J over the jazz. If you like P bass then P/J will give you more option, but if you don't like P bass then P/J setup might not cut it for you ......
First treat the P/J as a precision bass with a added J bridge pickup. Another word means I can use this bass as a stand alone P, solo the bridge pickup to get that bridge pickup jazz tone or combine both to get a different tone. On my P/J bass using both pickup together gives me abit of scoop tone but not as much as 2 jazz pickup together.