Epithet may have started being used as a genteelism for adjective as in the common locution racial epithets but it very quickly by association came to mean slur or derogatory expression. This is an American usage as there is no mention of it in the Shorter Oxford Dictionary. In Webster’s New Collegiate Dictionary (1979 edn.) there it was: a disparaging or abusive word or phrase. This was offered after the primary meaning which would be the classical: a characterizing word or phrase accompanying or occurring in place of the name of a person or thing. No examples were given but I offer chaste Diana and The Iron Lady as typical. A great many Americans would resist the idea that the latter was an epithet for Margaret Thatcher whom they profess to admire. Is Lionhearted an epithet in the secondary American sense? Hardly. Internal strife in the family of meanings of a word is to be deprecated.
Genteelism: By genteelism is here to be understood the substituting, for the ordinary natural word that first suggests itself to the mind, of a synonym that is thought to be less soiled by the lips of the common herd, less familiar, less plebeian, less vulgar, less improper, less apt to come unhandsomely betwixt the wind & our nobility.(from Modern English Usage by H.W. Fowler)