periodontist vs tooth doctor

periodontist and tooth doctor both are nouns.

Word	Noun	Adjective	Verb	Adverb
periodontist	Yes	No	No	No
tooth doctor	Yes	No	No	No

As nouns, tooth doctor is a hypernym of periodontist; that is, tooth doctor is a word with a broader meaning than periodontist:

periodontist: a dentist specializing in diseases of the gums and other structure surrounding the teeth
tooth doctor: a person qualified to practice dentistry

Other hypernyms of periodontist include dental practitioner, dentist.

periodontist (noun)	tooth doctor (noun)
a dentist specializing in diseases of the gums and other structure surrounding the teeth	a person qualified to practice dentistry