Difference between "public figure" and "celebrity?"

Ultimately, what makes a clearer distinction between the words "public figure" and "celebrity" or "famous?" Many people use them all synonymously, but the reality is that they aren't all equal.

People do not use these synonymously.

No one says 'public figure' at all except as a legal term, especially for those extended fewer privacy protections. It essentially means 'someone who has put themselves out there' and is thus fair game for public attacks. The precise definition varies by jurisdiction, though.

Celebrity is now the agent noun of famous, much moreso than its original sense of someone who is celebrated. Its synonym, though, is the noun phrase famous person, not the adjective famous.