My experience is more like what you said with the Cutlass, "The cutlass is a thicker tone that sounds rounder and less defined in the top end. Its really cool, but its going to come down to what you like." Caprice is rounder and thicker, but it doesn't have the same highs as my Jazz. It has a completely different character that I can't entirely put my finger on. Not really like any bass I ever owned before. Round and thick are definitely in the description though

, but not round and thick like an old Gibson

. Maybe round and thick like an old Gibson but without any of the muddiness, and a little more definition?
I hate describing bass sounds... I'm gonna quit it now. The Caprice is awesome. It just isn't a jazz bass, and if anything can get closer to a P sound, than a J sound.