I just bought my first one (MIJ fretless JB62 Jazz) last month for several reasons. For me it's fun to try my hand at a new skill that, while challenging, is also not a far departure from what I'm used to already. You've got the same strings in the same layout, but you can't just pick it up and play it without learning some things first.
I liken it to riding a fixed gear bicycle: for the most part it's just like any other bike that most people can just hop on and ride--except that, crucially, you cannot coast and that forces you to re-learn how to ride a bike. I've been playing bass for decades and now I am, in a way, re-learning how to play bass!
Somebody said something upthread about it keeping them out of a rut and I would agree. It's something different enough that is fun to jump to, and makes you appreciate what you're used to when you go back to that, with something new to bring to that table. It seems that the skills developed in learning fretless (left hand technique, a better ear) should translate positively to fretted basses as well. I'm already paying much more attention to what my left hand is doing than ever before.
Other important reasons: mwah, it looks stunning (unlined!), and of course... Jaco.