big gun vs big shot

big gun and big shot both are nouns.

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

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

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

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