big fish vs big cheese

big fish and big cheese both are nouns.

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

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

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

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