big cheese vs big gun

big cheese and big gun both are nouns.

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

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

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

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