They are a superset of the Fourier transform. The discrete Fourier transform is at one extreme of a continuum.
Wavelets are better suited to time-limited data.
Unfortunately, it does not exist. The mexican hat wavelet is not orthogonal. The first symmetrical and orthogonal wavelet discovered was a simple square wave, which is not continuous, and is known as the Harr wavelet. These are not unique, and more nearly symmetric continuous wavelets exist.