rage vs anger

rage and anger both are nouns.

Word	Noun	Adjective	Verb	Adverb
rage	Yes	No	Yes	No
anger	Yes	No	Yes	No

As nouns, anger is a hypernym of rage; that is, anger is a word with a broader meaning than rage:

rage: a feeling of intense anger
anger: a strong emotion; a feeling that is oriented toward some real or supposed grievance

Other hypernyms of rage include choler, ire.

As nouns, anger is a hypernym of rage; that is, anger is a word with a broader meaning than rage:

Other hypernyms of rage include angriness.

rage (noun)	anger (noun)
an interest followed with exaggerated zeal	belligerence aroused by a real or supposed wrong (personified as one of the deadly sins)
a feeling of intense anger	a strong emotion; a feeling that is oriented toward some real or supposed grievance
a state of extreme anger	the state of being angry
violent state of the elements
something that is desired intensely

rage (verb)	anger (verb)
feel intense anger	make angry
be violent; as of fires and storms	become angry
behave violently, as if in state of a great anger