orthodontia vs dental medicine

orthodontia and dental medicine both are nouns.

Word	Noun	Adjective	Verb	Adverb
orthodontia	Yes	No	No	No
dental medicine	Yes	No	No	No

As nouns, dental medicine is a hypernym of orthodontia; that is, dental medicine is a word with a broader meaning than orthodontia:

orthodontia: the branch of dentistry dealing with the prevention or correction of irregularities of the teeth
dental medicine: the branch of medicine dealing with the anatomy and development and diseases of the teeth

Other hypernyms of orthodontia include dentistry, odontology.

orthodontia (noun)	dental medicine (noun)
the branch of dentistry dealing with the prevention or correction of irregularities of the teeth	the branch of medicine dealing with the anatomy and development and diseases of the teeth