Why does FM sound better than AM?

Yes, phase noise exists, but I would think that in practice amplitude noise is greater.

In physics, when a wave passes from one medium to another, its frequency is supposed to stay the same. Even if this isn't perfectly true in the real world, I would think amplitude is more likely to decrease due to obstacles, distance, and the medium absorbing some energy.

If the noise is white gaussian noise (AWGN) then the phase noise is essentially the same as the amplitude noise (by the properties of the Fourier Transform).

Also, the information in AM is carried by the relative amplitude of the signal. Flat attenuation like you're describing doesn't really distort the AM signal. What does impact both AM and FM is frequency selectivity. Imagine light traveling through a prism and being split by frequency. If there are obstacles in the way, some colors won't pass through as well. The is can cause distortions in FM as the receiver loses lock on the signal. Am suffers from this too, but people are less likely to notice because they're used to these distortions -- these kind of effects happen with sound too.

As other posters have mentioned, the reason FM sounds better is that it has more bandwidth for the signal.

Very interesting, I'll have to look into AWGN and the Fourier transform. I guess in the trees blocking the flashlight example that's not at all AWGN.

Although while we care about the relative amplitudes in AM, AWGN would make this harder to pick out if the signal is attenuated. Is the same idea true for frequencies? I don't see a direct parallel here.

You do get some frequency deviation from AWGN. The sum of equal-amplitude 100Hz and 120Hz sine waves is a 110Hz sine wave that "beats", which is to say, is amplitude-modulated, at 10 Hz (or 20Hz from a certain point of view). So, if you have a 120Hz signal and you add a 100Hz signal to it, you should expect that to deviate the frequency of the detected signal downwards. AWGN will have varying, random amounts of all frequencies in it, which will cause varying, random amounts of frequency deviation as they add to your signal.

It's definitely easier to understand in the Fourier domain.

But the two are the same thing. You took the fourier transform of white noise and find white noise, both real (amplitude noise) as complex (phase noise).

You can think of it like this: the noise is not about the phase changing, it is about your ability to tell what the phase is. The noisier the signal gets, the harder time you will have to tell what the amplitude is, as well as what the phase is.