First of all, I'd just like to say I don't
exclusively use Jazzmasters (I have a couple of Rickenbackers, SG, LP and a lovely "reverse offset" Yamaha SGV800 amongst others.)
I've had my white Jazzmaster since '92. I just always thought they looked gorgeous, ever since I saw Robert Smith using one on an early Cure video. I've always like guitars whith a chimey kind of sound (hence the Rickenbackers) and the Jazzmaster is certainly very good for that. It also has the only tremolo system I can get on with (though I had to make a new arm to suit my playing style). I absolutely hate the Strat-type trem and find Bigsbys too cumbersome.
It was when I acquired the Jazzmaster that I first started to get into different tunings and it just seemed to deal with these tunings better than the other guitars I had. I now have two Jazzmasters and one is in open D and the other DADGAD.
For gentle strumming and picked parts it's my go-to guitar, along with the Rickenbacker, but for leads and heavy distortion I'll usually go for the SG or LP, though the Rick 330 sounds pretty amazing with distortion too, as it's much darker sounding than a lot of people realise. If I must have use of a trem with distortion, then I'll use the Jazzmaster, but it's times like this I feel like adding a Jazzmaster trem to an SG
Obviously the Jazzmaster will get you those Cure and Cocteaus tones, but it also comes pretty close to the Micro-Frets that Dave Fielding uses. In my last band, the other guitarist had a Micro-Frets and I thought it sat pretty much slap-bang between the Jazzmaster and Rick 330 soundwise. [ Incidentally, Mark Burgess told me that Reg really doesn't care what guitar he uses. It's just a tool to him!]
When it comes down to it, play what you like! I've always had an aversion to Strats myself, but I'm in a small minority there!