Power spectral density interpretation

After reading this question: PSD (Power spectral density) explanation I am still a little confused as to what extra information the PSD gives us over simply taking the magnitude of the fourier transform.

So my understanding is that the PSD is equal to:

$$|X(\omega)|^2$$

but what does that tell me that simply

$$|X(\omega)|$$

does not. I am assuming that squaring the fourier transform's magnitude somehow makes the number more physically (or mathematically) relevant? So how do I interpret the values of the power spectral density, especially when the signal is not a physical one and hence has no actual energy (say for example a financial signal or an image)?

Also, does the power spectral density contain information about the probability of the frequencies in the signal? As in is there a way to use the power spectral density to make a statement about the significance level of making a claim about a harmonic frequency in a stochastic signal? So something kind of like checking the significance of the coefficients from an AR model?