nut case vs oddball

nut case and oddball both are nouns.

Word	Noun	Adjective	Verb	Adverb
nut case	Yes	No	No	No
oddball	Yes	No	No	No

As nouns, oddball is a hypernym of nut case; that is, oddball is a word with a broader meaning than nut case:

Other hypernyms of nut case include eccentric, eccentric person, flake, geek.

nut case (noun)	oddball (noun)
a whimsically eccentric person	a person with an unusual or odd personality