Fatter = wider not deeper
Thinner= the opposite.
However is not quite that simplistic as there have thin neck Ps and twin PUP Ps and even active Ps.
As a genearlism though Pluckies post aint far wrong.
the P was teh 1st bass Leo desinged and mass produced (called Precision beacause it had frets, and prior to that [arguably] basses were fretless) 34" scale bass.
Once used to the idea of playing bass horizontally, quite a few artists wanted tonal variation and a faster neck so Leo designed teh J with 2 pups a different shape body and a more slender and narrower neck.