odium vs hatred

odium and hatred both are nouns.

Word	Noun	Adjective	Verb	Adverb
odium	Yes	No	No	No
hatred	Yes	No	No	No

As nouns, hatred is a hypernym of odium; that is, hatred is a word with a broader meaning than odium:

odium: hate coupled with disgust
hatred: the emotion of intense dislike; a feeling of dislike so strong that it demands action

Other hypernyms of odium include disgust, hate.

odium (noun)	hatred (noun)
hate coupled with disgust	the emotion of intense dislike; a feeling of dislike so strong that it demands action
state of disgrace resulting from detestable behavior