I reckon it's the other way around - a 'P' bass has a noticeable hump in the midrange, so it is more apparent in most mixes, the Jazz has an inbuilt 'mid-scoop' due to the phasing of the pickups. All of this can be altered/changed outboard with amps, effects, wang, EQ etc...
The big difference as a player is that the neck on a 'P' bass is a bit fatter, with the Jazz necks traditionally having a slimmer profile.