big shot vs big cheese

big shot and big cheese both are nouns.

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

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

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

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