Can STFT (Short-time Fourier Transform ) be more useful than FFT for analyzing stationary signals under some circumstances?
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.