rhyme vs clerihew

rhyme and clerihew both are nouns.

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

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

rhyme: a piece of poetry
clerihew: a witty satiric verse containing two rhymed couplets and mentioning a famous person

Other hyponyms of rhyme include doggerel, doggerel verse, jingle, limerick.

rhyme (noun)	clerihew (noun)
correspondence in the sounds of two or more lines (especially final sounds)	a witty satiric verse containing two rhymed couplets and mentioning a famous person
a piece of poetry

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