elementary school vs primary school

elementary school and primary school both are nouns.

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

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

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

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

elementary school (noun)	primary 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