orthodontics vs dental medicine

orthodontics and dental medicine both are nouns.

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

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

orthodontics: 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 orthodontics include dentistry, odontology.

orthodontics (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