Wavelet analysis is similar to FFT analysis in that it expresses a time series as a sum of functions. Because the functions are not simple sine waves, wavelets are able to better handle time series that have many spikes and are discontinuous. Because wavelets are localized in both time and frequency, they are able to handle large and noisy data streams much better than FFT, which is localized in frequency only.
For more details, refer to Wavelets and Filter Banks, by Gilbert Strang and Truong Nguyen (Wellesley ‘ Cambridge Press, 1996).