institution vs academy

institution and academy both are nouns.

Word	Noun	Adjective	Verb	Adverb
institution	Yes	No	No	No
academy	Yes	No	No	No

As nouns, academy is a hyponym of institution; that is, academy is a word with a more specific, narrower meaning than institution:

institution (noun)	academy (noun)
the act of starting something for the first time; introducing something new	a learned establishment for the advancement of knowledge
an establishment consisting of a building or complex of buildings where an organization for the promotion of some cause is situated	a secondary school (usually private)
a hospital for mentally incompetent or unbalanced person	a school for special training
a custom that for a long time has been an important feature of some group or society	an institution for the advancement of art or science or literature
an organization founded and united for a specific purpose