I can't add much to Lozz's detailed answer but to say both are good all rounders. They are used in soul, funk, jazz, country, indie, metal, blues etc where as some basses are more obviously genre specific (you can break the rules but I merely make the point). I too am a P-bass player, the Jazz seems to be, & I am hugely generalising here, for flashier playing or players with smaller hands.
The one bit from above I've never really got is when people say about Jazzes being easier to play. You get used to [b]any[/b] neck profile and ultimately if you are practising with one instrument you will bond with it anyway. I always feel a bit clusmsy on a Jazz but this is simply because I'm not used to them & certainly not a design fault.
Your best bet is to go into your local shop and try both. Take on board the comments made but go with the instrument YOU feel most comfortable with.