big shot vs big fish

big shot and big fish both are nouns.

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

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

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

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