Amplitude modulation uses the modulation of the amplitude of a carrier wave to transmit data while leaving the carrier frequency constant, and in the case of FM it's just the other way round.
Why is a modulation of the frequency in an AM signal not used to transmit additional information, e.g. two audio channels or RDS information?
It turns out that we DO simultaneously use frequency and amplitude modulation: our everyday term for this is "single sideband." As described on Wikipedia, single-sideband is a special case of quadrature modulation in which the quadrature component (Q) is derived directly from the in-phase component (I) of the modulating signal.
This effect becomes evident upon examination of the waveform of an SSB signal produced by modulating tones of 700Hz and 1900Hz:
Note that the period of the envelope is 833$\mu$s, a frequency of 1200Hz, which is the difference between the frequencies of the two modulating tones. At the "cusp" of the waveform, it experiences a 180-degree phase reversal, whose accurate reproduction is essential to a clean output. The mathematical combination of these amplitude and phase modulation components produces the two tones that are offset from the frequency of the carrier wave.
The two different techniques are not just different ways of modulating the signal. As you can see from the frequency units in your examples, FM carrier frequency is about two orders of magnitude higher than AM.
This means that FM can modulate a much higher signal frequency than AM, and this leads to greater fidelity (i.e. sound quality).
The FM carrier frequency is also a lot less susceptible to interference effects (particularly atmospheric). This is because it is fairly easy for random attenuation, reflection and refraction to change the amplitude of a signal (thus interfering with AM) but more difficult for it to change the frequency.
When FM became popular in the 1970s, it was regarded as a major technology leap - similar to DAB today. So much so, that Steely Dan wrote a song about it.
