major league vs big league

major league and big league both are nouns.

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

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

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

Other synonyms of major league include majors.

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