What is the difference between the PSD of a deterministic and stochastic signal?
Generally, one cannot compute the Fourier transform of any random process, because of integrability reasons. The PSD is one way to define it, based on autocorrelation. 

And now, this mathematically sound definition of a periodogram is consistent in some way: if you have a realization of a random process, on a limited time interval, then its Fourier transform can be computed, and is consistent with the PSD, which you'd expect.