the nature of STFT is to be applied on non-stationary signals. for stationary signals STFT and FFT sounds exactly same to me. However, I was wondering if STFT can lead to better results compared to FFT (when applied on stationary signals), especially when optimized the STFT window function or other possible parameters?
In the continuous version of the short-time Fourier transform, if you choose a uniform unit-height window, it just boils down to the continuous Fourier transform.
In practice, for the discrete case, you can find STFT parameters that just give you the same information as that of the FFT. Hence, STFT can always be as useful as the FFT.
Since real signals are often finite length, they are not stationary in the purest sense. Even if the noise is "stationary", we only have access to few realizations.
For instance, the estimation of the parameter of an additive noise from a few realizations would probably be more robust in the STFT space than from the FFT.