big gun vs big deal

big gun and big deal both are nouns.

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

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

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

big gun (noun)	big deal (noun)
an important influential person	anything of great importance or consequence
	an important influential person