Given a sound clip (whose source is an FM radio signal) I'd like to measure how different from background noise (the "in-between channels" noise) the sound clip is, so that I can compare the "non-noisiness" (and thus the quality) of two radio signals. Is there a good way to do this?
If the noise is (close to) stationary, you could use one of the many noise-floor estimation methods to get an estimate of the noise power and, consequently, of the SNR. Search for "noise floor estimation" and "noise floor tracking". You'll find a lot of publications and patents.
@Matt's answer is nice and general (+1), but I would also look at the noise level in silent passages on a tuned station. Since there is a carrier frequency, this might be different from noise levels measured "in-between channels" where there is no carrier, as you suggest.
Lastly, this might not be what you are asking about, but since your source is FM radio, which presumably contains a lot of music, you could find a pristine (i.e. uncompressed) copy of a song playing on the radio, time-align the signals from the radio and the reference song, and compare the spectra. The difference would be noise.
There is a problem with this approach, however, and that is the fact that many radio stations equalize and apply dynamic range compression and other effects to the music they play. This will lead to differences in the reference and recorded spectra not created by noise. Perhaps public or college radio would not mess with the broadcast signal, but that is just a guess.