I have played my '87 Jazz Bass Special (P/J) mostly on the P pup for 25 years, as well as my '01 MIA Jazz for the last ten years. When comparing them side-by-side I find that the Jazz front pup can indeed sound similar to the P, but doesn't have the same punch, just a similar tone. That's to be expected. As for preference, I never can decide. Both sound wonderful to me. The Jazz is "smoother," whatever that means. The only reason I might prefer a straight-out Jazz Bass over a straight-out Precision has more to do with the neck shape. I prefer the Jazz's narrower nut width. Fortunately, with my Jazz Bass Special I have the best of both worlds.
On another note, I also like the sound of either bass with both pups on full, depending on the song. While the tone is a hair scooped that way it actually seems to still cut through in a mix. If I get another bass I'd like to try a Blacktop Jazz.