periodontics vs dental medicine

periodontics and dental medicine both are nouns.

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

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

periodontics: the branch of dentistry dealing with diseases of the gums and other structures around the teeth
dental medicine: the branch of medicine dealing with the anatomy and development and diseases of the teeth

Other hypernyms of periodontics include dentistry, odontology.

periodontics (noun)	dental medicine (noun)
the branch of dentistry dealing with diseases of the gums and other structures around the teeth	the branch of medicine dealing with the anatomy and development and diseases of the teeth