If we already had Short-time Fourier transform for better analysis of a signal than Discrete Fourier Transform, then what was the need that leads to development of Wavelet 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.