How to Quantify a Difference in Spectral Information of two Signals?
Quantifying a difference in the frequency domain can be useful if the information on how your signals differ is better expressed in the Fourier space. If this assertion is valid in your case, hereafter are hints on methodology, and basically the sort of you can find in Shazam to distinguish music from a short listening period.
By the way, I think that I have been trying to find an answer to a related question  in 50% of my research hours pertaining to spectral or harmonic  analysis (the rest was coding). I am not fully successful yet.  I would however rephrase it as:

How could I easily find quantitative or selective (Fourier) features for
relatively stationary signals?

By quantitative I mean that the feature set should be able to measure scalar values related (in a monotonic way) to a quantity I seek in the original signal. By separative I mean that features that could tell whether signals are &quot;apart&quot;. Those two concepts could be related by proper distances or metrics between the feature sets of two signals.
In your proposal, there is a notion of location in the frequency axis, a sort of median of spectrum amplitudes : 50% of the area under the curve for the FFT. A notion of amplitude is skipped, as you may normalize spectra. There is no notion of uncertainty or spread either. I am often trying to consider, as first, feature  triplets of Location-Amplitude-Spread ($f^k_0,f^k_1,f^k_2$) to describe the morphology of signals, spectra, or other suitable sparsifying transformations on signals.

Those three parameters could be compared to higher-order cumulants or  moments of a distributions. They are related to integrals of the shape:
$$M_d=∫^∞_{−∞}(t−c)df(t)dt$$
or their ratios. A related question is Finding the right measure to compare sound signals in the frequency domain.