In my experience with various basses I've owned, I get a 'different' kind of bottom from a P... ...
Furthermore, I think if you favour the P on the PJ, you get a more focussed sound (more low mids) as when you do the same on a JJ (more low end).
One crucial difference might be that with the PJ, a J SC pickup doesn't really make sense (hum cancelling wise), whereas you can easily use a couple of SCs (especially Sadowsky JJs) on the JJ.
In my experience I always favoured Sadowsky SCs on my 5 strings.
But apparently with the Sadowsky 4 strings the difference between SCs and HC are minimal.... But that's a different discussion