madness vs insanity

madness and insanity both are nouns.

Word	Noun	Adjective	Verb	Adverb
madness	Yes	No	No	No
insanity	Yes	No	No	No

As nouns, insanity is a hypernym of madness; that is, insanity is a word with a broader meaning than madness:

madness (noun)	insanity (noun)
unrestrained excitement or enthusiasm	relatively permanent disorder of the mind
the quality of being rash and foolish
a feeling of intense anger
obsolete terms for legal insanity
an acute viral disease of the nervous system of warm-blooded animals (usually transmitted by the bite of a rabid animal); rabies is fatal if the virus reaches the brain