poem vs rhyme

poem and rhyme both are nouns.

Word	Noun	Adjective	Verb	Adverb
poem	Yes	No	No	No
rhyme	Yes	No	Yes	No

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

poem (noun)	rhyme (noun)
a composition written in metrical feet forming rhythmical lines	correspondence in the sounds of two or more lines (especially final sounds)
	a piece of poetry

poem (verb)	rhyme (verb)
	compose rhymes
	be similar in sound, especially with respect to the last syllable