Precision Bass: traditionally a wider, flatter neck.
Jazz Bass: traditionally a more narrow neck (smaller nut width)
Precision Bass: full, deep, fundamental sound; punchy, not too bright; great choice for more traditional rock styles, heavier stuff, country, 1960s R&B - anything where the bass should lay a grooving foundation, but should not be too prominent.
Jazz Bass: neck pickup: not as deep and punchy as the P's split coil pickup, tighter, more focussed, but also thinner; bridge pickup: singing mids and bright treble, not too much bass; both sounds can be blended; great choice for music where the bass plays a more central role and solos sometimes (jazz, funk, modern R&B), great for modern rock styles where the bass plays a more prominent, melodic role; great for bigger ensembles with lots of instruments, where the tighter, more focussed sounds sits better in the mix and does not muddy up the lower frequencies.
I personally also like the PJ-combo: you get the deep & punchy P-bass sound, and you get the "goodies" from the J-bass, you can dial in the brighter, singing, mid-focussed bridge pickup that will help cut through the mix when you want to get funky or play a solo.