anthem vs paean

anthem and paean both are nouns.

Word	Noun	Adjective	Verb	Adverb
anthem	Yes	No	No	No
paean	Yes	No	No	No

As nouns, paean is a hyponym of anthem; that is, paean is a word with a more specific, narrower meaning than anthem:

anthem: a song of praise (to God or to a saint or to a nation)
paean: (ancient Greece) a hymn of praise (especially one sung in ancient Greece to invoke or thank a deity)

Other hyponyms of anthem include dithyramb, doxology, choral, chorale, canticle, hymeneal, pean, recessional.

anthem (noun)	paean (noun)
a song of praise (to God or to a saint or to a nation)	a formal expression of praise
a song of devotion or loyalty (as to a nation or school)	(ancient Greece) a hymn of praise (especially one sung in ancient Greece to invoke or thank a deity)