senior high vs secondary school

senior high and secondary school both are nouns.

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

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

senior high: 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 senior high include Gymnasium, lycee, lyceum, middle school.

senior high (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