By power spectrum we are talking about an ASD (Auto Spectral Density) or PSD (Power Spectral Density). In fact PSD is often used when really ASD would be more correct.
Taking any signal and performing a frequency analysis using an ASD or a PSD will give the energy over a range of frequencies.
In vibration analysis, this data is used for resonance (or natural frequency) analysis. If a structure or component has a resonance at a particular frequency that is being excited by normal conditions then the part needs to be redesigned. This means either changing the mass or the stiffness to move the natural frequencies to another, higher frequency. The Auto Power Spectrum will show this.
When studying acoustics or vibration often transfer functions are used. These are usually, but not always, based on the auto power spectrum.
Studying transfer functions is very useful whether you are interested in noise, vibration or both. How much will this new damper reduce the vibration? How much will this new window reduce the noise from the building site? All these and more are analyses based on the measured power in the spectrum.