What is the main difference between the "Appeal to Popularity" and the "Appeal to Common Belief?"

Not much of a difference when both refer to a belief. But the appeal to popularity is more generic that can be applied to ideas, products, and other non-beliefs. For example,

The new Tickle-Me-Buzz-Lightyear toy cannot stay on the shelves, therefore, it is a great toy!

This more accurately is about us reasoning that because a toy sell well (is popular), that it must be good, therefore appeal to popularity is a better fit. However, if we said

Parents everywhere believe that the Tickle-Me-Buzz-Lightyear toy is the best toy ever, therefore, it is a great toy!

This is would be common belief.

Not much of a difference, and it really does not matter that much. Informal fallacy names are just a way to help us identify the problematic reasoning. You can call it "fallacy of stupidity" and it wouldn't really matter, as long as you realized that just because something is popular or common, does not make it good/right/true.