big shot vs big gun

big shot and big gun both are nouns.

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

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

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

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