Why Wavelet developed when we already had Short-time Fourier transform
The short-time Fourier transform doesn’t offer better analysis of data than the discrete Fourier transform, it offers a different kind of analysis.  The DFT offers an exact decomposition of data to a frequency representation. The STFT offers an approximate decomposition to a time/frequency representation.  Which is better depends on what you are after.  The same holds true of the Wavelet transform.  Wavelet transforms can be thought of as decomposition to a time/frequency representation, but wavelet transforms generalize the concept of decomposition.  Different wavelet functions have been created so you can choose a decomposition that suits your needs.