Epitaph, Epithet

An epitaph is a tribute inscribed on a tombstone in honor of the person buried there.

An epithet, unlike an epitaph, is often an insult based on race, class, religion, politics, etc.: The mob was shouting racial epithets.

Otherwise, an epithet is a kind of nickname. It is a word or brief phrase that illustrates a defining trait of someone or something: Alexander the Great, the wine-dark sea.