A FT provides enough information to perfectly reconstruct a bandlimited input, including exact localization of everything. But the information about localized behavior is spread out in the FT phase across an infinite extent of the frequency domain.
However, in practice, FFT analysis is done using one fixed size of window, without reconstruction, and looking only at local magnitudes, which do not contain enough info about whether an impulse (tone burst, etc.) occurred in the left, right or middle (etc.) of the FFT aperture.
Whereas, wavelet analysis is more often done using various sizes and overlaps of basis functions. If an impulse (narrow-band burst, etc.) is located at an unknown position within one wavelet, another set of smaller and offset wavelets might identify in it just one, and thus better localize it.
One could also do something similar by using a bunch of sets of different sizes of windowed FFTs at varying overlaps per set the get a roughly equivalent quality of localization information, but at a likely much higher computational cost than using wavelets for analysis.