madness vs wrath

madness and wrath both are nouns.

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

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

Other hyponyms of madness include lividity.

madness (noun)	wrath (noun)
unrestrained excitement or enthusiasm	belligerence aroused by a real or supposed wrong (personified as one of the deadly sins)
the quality of being rash and foolish	intense anger (usually on an epic scale)
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