The Fourier has the disadvantage that you can't arrange the components into a time hierarchy; that is, no component occurs "before" any other.
The Wavelet transform _does_ have a natural time hierarchy. This makes it much better for streaming compression like voice calls.
The Fourier perfectly describes signals of infinite duration (think tone or color) while the Wavelet perfectly describes the position of things within a signal (think rhythm or space).
With the Fourier filtering is really easy. You can do hard, hard cutoffs -- literally no contributions within a certain frequency band -- just by removing components of the decomposition. Similarly, you can accurately apply any arbitrary mathematical filtering function.
The disadvantage of the Wavelet is that, well, the only meaningful transformation you can apply to it is compression -- dropping the shorter timescale components. If you want to filter, it's not enough to trim off timescale components because the wavelet itself can contain any frequency components. There's also nothing like a simple mathematical function you can apply to the coefficients to get a smooth filter.