i think both basses are in their own class.
for LP basses their tone inclines more towards the woofy deep, resonating tone. DC Talk fans will know that Otto Price uses a LP bass on their records.
Jazz basses are the bread and butter for most bass players, their tonal range is huge thus finding it's place in the arsenal of most bass players.
i think if you would give me a choice, i'd go for jazz basses, not saying that LP basses are lousy, but just a preference. Some others may choose the LP over the jazz and i think that's equally cool.
at the end of the day, it's not the bass that defines the bass player but the bass player that defines the bass.