Wavelet transform is normally applicable for non stationary signals i.e.,the spectral content of the signal changes with time.Other transforms especially Fourier transform is applicable for stationary signal.In Wavelet transform multiresolution is possible so that the same signal can be seen with different resolutions(like we are using multiple lenses to view the signal)

There are differences in techniques and in the family of wavelet chosen. As for the family of wavelet (shape), it all depends on the features of your signal you are interested in.