[jmts]

FM works better because it is easier to detect the change in frequency independently of any change in the amplitude.

I'm unsure of what the correct terminology would be, but (for my linear algebra brain) you could say something like, for FM the noise dimension is orthogonal to the signal dimension, while for AM the noise and signal dimensions are the same. Therefore for FM any change in amplitude in the noise dimension should be mostly isolated from the signal dimension, while it is essentially impossible to tell what is noise and what is signal for AM - you could probably do some radio equivalent of a differential pair in order to detect noise and remove it, but then why would you bother when FM has improved noise rejection anyway.