Most explanations of classes vs sets motivate the discussion via Russell's paradox. This feels like a kind of "gotcha technicality" to me, in the sense that one might reasonably prove mathematical theorems not specifically contrived to trip one up involving big class-type things using the word "set" everywhere, it will "just work", and one can go back and clean up the language later.
Are there other important implications of the class set distinction? i.e., since the class vs set distinction is a "limiting" construction (it specifically prevents certain statements that are allowed in naive set theory), are there any other "big" theorems that are provable in naive set theory but not provable in class/set theory?
What are some important metamathematical implications of the class/set distinction?
Cantor’s theorem which implies that there is no bijection between a set and it’s power set is false for the class of all sets.
Another example is that if the class of all sets were a set then by Zorn’s lemma there has to be a maximal element, a contradiction.