There are two different things involved here. First is the type of signal encoding itself, which is amplitude modulation (AM) and frequency modulation (FM). The second thing is the bandwidth allocations set up by the FCC.
As far as the modulation methods themselves, either can be used to create an accurate reproduction of the sound, at least as far as the human ear can hear. The major disadvantage of AM is that whenever two signals of the same frequency are mixed together, their amplitudes are simply added together, so that it becomes impossible for an AM receiver to discriminate one signal from the other. The effect of this is that any noise at all on the same frequency cannot be removed from AM broadcast signals by the receiver. FM, since it is frequency modulated, just tracks the frequency of the loudest signal. Quieter signals on similar frequencies are ignored by the receiver. This means that FM signals don’t get all that annoying static that AM signals are prone to.
The typical human hearing range roughly 20 Hz to 20 kHz, although some people can hear as high as 24 or 25 kHz (rare). This range tends to decrease with age, so older people rarely hear above 15 kHz. This is where the FCC limits start to come into play. For an AM radio station, how much bandwidth they allow translates into how much radio bandwidth they take up. I hope I’m remembering these numbers correctly, but I believe that AM radio is limited to a single channel (which means no stereo music) with a bandwidth of only 10 kHz. Since this is well below the range of what most people can hear, there is audible signal loss in AM radio. FM does slightly better, with a bandwidth of 15 kHz, but it is still below the 20 kHz that many people can hear.
An MP3 that is “radio quality” will be comperable to an FM broadcast radio signal.
AM and FM are both analog systems, which means that any noise present in all of the pre-amplifiers and signal conditioners, transmitters, receivers, etc. all gets mixed into the final sound and causes noise. Digital systems, just by their nature, are 100 percent faithfully reproduced within the limitations of their sampling rate, due to the fact that any noise that is not large enough to flip a bit is completely ignored by all components throughout the system. So, FM is better than AM, and digital is better than either one.
Note - if the sampling rate of a digital system is too low, it can still produce audible distortion of the signal. A good example of this is phone systems, which are very narrowly limited in bandwidth before being digitized in order to maximize the number of voice channels that can be carried through the system. Digital phone systems tend to have roughly the same signal quality as AM radio as a result of this.