Power spectral density interpretation
it doesn't tell you new information, but because the abs value function is non-analytic (i.e. not all of its derivatives are continuous), and the magnitude-squared is, then the latter can be manipulated mathematically in ways that the former cannot.  one important property of the power spectrum is that it is the Fourier transform of the autocorrelation function in the time domain.

the PSD gives you the energy of each frequency component for a deterministic signal and the expectation value of the energy for each frequency component for a random (or "stochastic" if you like jargon) signal.