An epitaph is written on a tombstone. An epithet is a nickname or a description of someone. Halloween graves often combine them: "Here lies Fearsome Frank, who bet that he could rob a bank."
Epitaph is usually the words inscribed on the stone, but it can also be a memorial statement about someone who has died. Epitaphs are usually grave (ha ha), but old ones can sometimes be unintentionally funny, like this one: "Here lies Lester Moore/Four slugs/From a forty-four/No less/No more." Here are some other examples of epitaph:
"Here lies one whose name was writ in water," is the epitaph he composed for his grave, in Rome. (New York Times)
Her epitaph, being written in brass instead of marble, has escaped the wear and tear of nearly three centuries. (Hunter Joseph)
He has picked out a cemetery plot, selected his tombstone and written his epitaph. (New York Times)
An epithet is a description of someone, often a nickname, like if you're tall and people call you Daddy Long Legs. It's not necessarily an insult, but these days it's used that way a lot, like a racial or sexist slur. It's the kind of thing people sling at each other, like "red headed stepchild." An epithet can be negative, but it doesn't have to be:
According to legend, it is the Golden State — an epithet that might originate from the discovery of gold in California in 1848. (The Independent)
Elizabeth Olsen still needs to be described as "Mary-Kate and Ashley's younger sister," but any day now she might shake that epithet. (New York Magazine)
Be wary if someone writes your epitaph, after all, it'll be inscribed on your grave. But don't be afraid of an epithet, or nickname, maybe they call you The Gorgeous Successful Person.