What is the difference between concept class and hypothesis
If one requires that $H = C$, then this is called the "proper PAC" framework compared to "PAC prediction" where we don't care about the representation of $h$ as long as the prediction error is small enough (i.e. we allow $H$ to be the class of all time-polynomial programs).

You can think of a concept as the set of inputs that produce the same label (e.g. all the images that show a chair form the concept "chair" or all the points in the same half space form the concept "true/false").