elementary school vs grammar school

elementary school and grammar school both are nouns.

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

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

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

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

elementary school (noun)	grammar 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
	a secondary school emphasizing Latin and Greek in preparation for college