Bite, gnaw, champ, gnash are comparable when they mean to attack with or as if with the teeth.
Bite fundamentally implies a getting of the teeth, especially the front teeth, into something so as to grip, pierce, or tear off.
Sometimes bite denotes to wound by biting.
In extended use bite implies unusual power of penetration or power of cutting into something so that it stings or pricks, or gives support to a good grip or hold.
Gnaw, on the other hand, implies an effort to bite something hard or tough; it implies repeated action and a slow wearing away, sometimes stressing one in preference to the other.
Therefore gnaw is used of what eats, frets, or corrodes something that is strong, resistant, or not easily affected.
Champ implies vigorous and noisy action of the teeth and jaws as they attempt to penetrate something hard or, sometimes, inedible. The word usually is associated with animals (as horses) and connotes impatience or extreme hunger, but it is also used of men who avidly apply themselves to the task of biting with their teeth and crushing with their jaws: often it suggests the flow and foaming of saliva.
Gnash usually implies the striking of the teeth against each other or a grinding of them (as in anguish, despair, or extreme rage); it often emphasizes this action as the visible sign of an overpowering emotion or distress.
Sometimes, however, it implies a savage biting that rends a thing in two or tears it apart.