big cheese vs big fish

big cheese and big fish both are nouns.

Word	Noun	Adjective	Verb	Adverb
big cheese	Yes	No	No	No
big fish	Yes	No	No	No

As nouns, big cheese and big fish are synonyms defined as:

Other synonyms of big cheese include big deal, big enchilada, big gun, big shot, big wheel, head honcho.

big cheese (noun)	big fish (noun)
an important influential person	an important influential person