As @Marcus Müller mentions a stochastic process does not have a Fourier transform.
From a practical point of view, assume you have $N$ realizations of a signal, either deterministic or stochastic. Then, in order to estimate the power spectral density, it is common practice to average the power spectral densities of the individual realizations. For both deterministic and stochastic signals this will result in a more accurate estimate of the power spectral density as $N$ increases.
On the other hand, if you try the same with the Fourier transforms of the individual realizations this will only work for deterministic signals. For stochastic signals the phase of the Fourier transform is random and averaging generally result in a spectrum converging to zero as $N$ increases.