governor's race vs campaign for governor

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

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

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

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

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