high school vs secondary school

high school and secondary school both are nouns.

Word	Noun	Adjective	Verb	Adverb
high school	Yes	No	No	No
secondary school	Yes	No	No	No

As nouns, secondary school is a hypernym of high school; that is, secondary school is a word with a broader meaning than high school:

high school: a public secondary school usually including grades 9 through 12
secondary school: a school for students intermediate between elementary school and college; usually grades 9 to 12

Other hypernyms of high school include Gymnasium, lycee, lyceum, middle school.

high school (noun)	secondary school (noun)
a public secondary school usually including grades 9 through 12	a school for students intermediate between elementary school and college; usually grades 9 to 12