A proverb is a short saying like "There are two sides to every story" or "Many hands make light work". These sayings are nice for grandmothers who want to give advice, but they are not appropriate for academic writing! Click here to see some more proverbs - but don't learn them!
A cliché is an overused phrase like "in a nutshell" (which means "in summary" or "summed up briefly"). This website explains more about clichés and gives some more examples.
PS. Sorry for joking about grandmothers :)
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