Bass or guitar?
In general, the idea behind the pickups is that they will add less colouration than regular pickups. So there is probably less variation in the sound that the different pickups put out, compared to the many variations that are on offer from companies like DiMarzio or Seymour Duncan (to name but two).
That is not to say that there aren't any differences, e.g. nothing sounds like the single coils in the Series instruments.
AXYs and MXYs are identical except for the shell size. Probably most basses have a pair of these. They sound transparent - for lack of a better word: if the neck pickup sounds different from the bridge pickup it's because the string sounds different between the two spots. If you swap the pickups the instrument will sound the same, so the sound is not so much in the pickups.
Compared to AXYs/MXYs, FatBoys have a bit more oomph in the lower regions, and not as much sparkle.
Don't know about the Ps and Js, but they have much narrower magnets and coils, and I guess a P will sound different from a J. Also the J is relatively weak sounding (which is where the output level trimpots come in handy).