First, to clear a common misconception: RAW does not provide inherently better quality. Professional photographers, when presented with printed versions of RAW and JPEG photographs, often cannot even tell the difference!
However, RAW does provide you with a more “pure" version of what the image sensor originally detected when the photo was shot, which makes it easier to edit later on with less possible degradation of the image. Thus, if you're planning on doing high levels of editing, for instance for HDR photography or photo manipulation purposes, RAW is probably the better format.
RAW files are also much larger than JPEG, which becomes problematic with photographers who tend to be very shutter happy, go long periods of time without downloading their pictures (such as while traveling), or simply do not have a large memory card. This can be alleviated somewhat by carrying extra memory cards, though this requires significant investment and organization.
A serious disadvantage of RAW is that it is a proprietary format, that is, individual to the camera vendor. For the user, this means that not only can you not easily exchange RAW files with other people, but you also need special software installed to first read and then process the file. While this doesn't particularly discriminate against any one type of user, it is a nuisance for many.
This has been somewhat alleviated with increasing use of the DNG file format, essentially a universalized version of the RAW format that you can convert to. However, many cameras, particularly those by Canon and Nikon, do not natively support DNG, which somewhat undermines its convenience. Keep in mind also that RAW converters are continually getting better, so that issues of time and convenience are becoming less and less so.
Because they require so much extra processing, they take longer to shoot, and are slower to work with on your computer if speed is an issue. Thus, using them for action shots, especially at higher resolutions, generally doesn't work too well except on the more high end cameras.