big gun vs big cheese

big gun and big cheese both are nouns.

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

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

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

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