The difference is pretty simple, noise is an extra signal not related to the original one, distortion is a change in the original signal.
For example:
Noise - you are trying to talk to your mother, your little sister is shouting in the background - that's noise.
Distortion - you are still trying to talk to your mother, your little sister puts her hand over your mouth, although you can still talk it's very hard to understand - that's distortion.