Granted, the JJ bass that I have has single coil pickups, and the PJ that I have is "reverse" PJ in that the two P pickups are inverted. So, my comparisons are probably the extremes of the discussion.
The JJ bass has very warm, open tone to it. The mid presence is nice and strong. It is one of the most "musical" sounding basses I have ever heard. The high treble is soft and subtle. The low end is a bit thinner compared to the PJ and not nearly as punchy. A lot of this is likely attributed to the single coils. Some JJs have the Sadowsky hum cancelling pickups, which are said to have a more modern tone.
The PJ, in its neutral position, is very modern sounding, and yes it does have a hum cancelling pickup in the bridge slot. The tone isn't really that "scooped" to my ear, but is much more flat than the JJ. The bottom end is much tighter. The B string on that bass is VERY quick and articulate. Plently of low end but focused. This may be attributed somewhat to the reversed position of the neck pickups. The B string definitely sounds better on this bass. On the other hand, the high end can be a bit more shrill and has to be tamed a bit.
The PJ also seems to offer a bit more variety in tones than does the JJ. As awesome as it is, the JJ sounds like a single coil jazz bass. The PJ does the modern thing, but using the passive tone control and the PU pan, you can get a lot of different textures out of it. For example bypassing the preamp and soloing the neck PU and you have very respectable old school P. In a pinch, you can also get the "J-burp" out of it. I am not much of a slapper, but the slap tone on this bass is amazing.
I haven't owned either of them very long, but in general I tend to think of the JJ as the more "traditional" tone and the PJ as the more "modern". Although, they are both still very much Sadowsky. The voice of the Sadowsky preamp is unique to itself.