rhyme vs poem

rhyme and poem both are nouns.

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

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

Other hypernyms of rhyme include verse form.

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

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