melic verse vs poem

melic verse and poem both are nouns.

Word	Noun	Adjective	Verb	Adverb
melic verse	Yes	No	No	No
poem	Yes	No	No	No

As nouns, poem is a hypernym of melic verse; that is, poem is a word with a broader meaning than melic verse:

melic verse: verse that is intended to be sung, especially Greek lyric verse of the seventh to fifth century B.C.
poem: a composition written in metrical feet forming rhythmical lines

Other hypernyms of melic verse include verse form.

melic verse (noun)	poem (noun)
verse that is intended to be sung, especially Greek lyric verse of the seventh to fifth century B.C.	a composition written in metrical feet forming rhythmical lines