elementary school vs grade school

elementary school and grade school both are nouns.

Word	Noun	Adjective	Verb	Adverb
elementary school	Yes	No	No	No
grade school	Yes	No	No	No

As nouns, elementary school and grade school are synonyms defined as:

elementary school and grade school: a school for young children; usually the first 6 or 8 grades

Other synonyms of elementary school include grammar school, primary school.

elementary school (noun)	grade school (noun)
a school for young children; usually the first 6 or 8 grades	a school for young children; usually the first 6 or 8 grades