The Fourier transform converts a signal into a continuous series of sine waves, each of which is of constant frequency and amplitude and of infinite duration. In contrast, most real-world signals (such as music or images) have a finite duration and abrupt changes in frequency.
Wavelet transforms convert a signal into a series of wavelets. In theory, signals processed by the wavelet transform can be stored more efficiently than ones processed by Fourier transform. Wavelets can also be constructed with rough edges, to better approximate real-world signals.
For example, the United States Federal Bureau of Investigation found that Fourier transforms proved inefficient for approximating the whorls of fingerprints but a wavelet transform resulted in crisper reconstructed images.
["Ten Lectures on Wavelets", Ingrid Daubechies].