car race vs auto race

car race and auto race both are nouns.

Word	Noun	Adjective	Verb	Adverb
car race	Yes	No	No	No
auto race	Yes	No	No	No

As nouns, car race and auto race are synonyms defined as:

car race and auto race: a race between (usually high-performance) automobiles

Other synonyms of car race include automobile race.

car race (noun)	auto race (noun)
a race between (usually high-performance) automobiles	a race between (usually high-performance) automobiles