three-year-old horse vs three year old

three-year-old horse and three year old both are nouns.

Word	Noun	Adjective	Verb	Adverb
three-year-old horse	Yes	No	No	No
three year old	Yes	No	No	No

As nouns, three-year-old horse and three year old are synonyms defined as:

three-year-old horse and three year old: a racehorse that is three years old

three-year-old horse (noun)	three year old (noun)
a racehorse that is three years old	a racehorse that is three years old