campaign for governor vs governor's race

campaign for governor and governor's race both are nouns.

Word	Noun	Adjective	Verb	Adverb
campaign for governor	Yes	No	No	No
governor's race	Yes	No	No	No

As nouns, campaign for governor and governor's race are synonyms defined as:

campaign for governor and governor's race: a race for election to the governorship

campaign for governor (noun)	governor's race (noun)
a race for election to the governorship	a race for election to the governorship