big league vs major league

big league and major league both are nouns.

Word	Noun	Adjective	Verb	Adverb
big league	Yes	No	No	No
major league	Yes	No	No	No

As nouns, big league and major league are synonyms defined as:

big league and major league: the most important league in any sport (especially baseball)

Other synonyms of big league include majors.

big league (noun)	major league (noun)
the most important league in any sport (especially baseball)	the most important league in any sport (especially baseball)