From a mathematical view, they have certain theoretical advantages. We'll examine here the practical differences that might affect the graphic arts. All commercial implementations of JPEG are restricted to RGB and gray-scale images. In contrast, all of the wavelet implementations we've seen can handle CMYK and multi-channel images. Lossless or lossy.
Wavelets allow true lossless compression.
It also appears that for moderate compression, wavelet artifacts are less objectionable than JPEG artifacts. Of course, under severe compression, neither technique is kind to a picture's contents. At this time, it appears that the biggest drawback for wavelet compression is simply that it is new and not yet widely supported. There is no public standard for image portability, and each vendor uses its own file format. Another consequence of novelty is that obscure patents may have been awarded in various countries, waiting to snare the unwary product developer.