Vaulted backs were common on high-end instruments of the 17th century, apparently. Some of them are pretty spectacular! Acoustically a vaulted back will be more 'reflective' and less 'active' than the usual 'flat' back. It's a little hard to pinpoint the exact acoustic consequences: I'd expect some differences in 'projection' and 'tone color'.
I believe the high end ones were part of the inspiration for the composite backs on 'Ovation' guitars, but, of course, those were not built to high end standards. To me they lack a wide tonal palette, and I attribute at least some of that to the vaulted back, which simply cannot vibrate much at low frequencies, which helps to color the sound. There is generally so much else wrong with those that it's difficult to assign causation to any one feature in particular. I've re-topped a few, and they always improve, but still tend to lack something in the 'color', OMO.