Is it because noise manifests itself in the transmitted signal's amplitude?
Looking through some literature shows that a side by side comparison of FSK with ASK does not give FSK a performance advantage over ASK in the presence of noise (1); however you can do some nonlinear processing with FSK in the demodulator that does produce a signal that is more tolerant to noise than ASK.
In some instances of FSK modulation/demodulation schemes, you can force the receive signal to clip (i.e. put the signal through a Schmidt trigger) so the amplitude noise is essentially eliminated. You can't do this with the ASK signal because this eliminates the information in the modulated signal. For the FSK signal, the clipped signal still carries frequency information with a slight phase error due to the noise signal. There is a threshold at which noise can cause enough phase error to affect the demodulation even after clipping the signal, but below that threshold, the signal is largely immune to noise.
FSK is just a modulation scheme and is not necessarily more immune to noise than ASK.
However, FSK can behave like a differential signal and, thus, FSK demodulators can exploit this and thus become more robust to noise. However if and only if that noise manipulates your positive and negative deviated frequency. In practise this means that the lower your frequency deviation the more robust you are against noise.
If you have trouble to understand why FSK can be similar to differential signals, think of each deviated frequency as an own signal. At every time there is exactly one frequency used. Thus you have a differential signal.