The difference is semantic.
The two degrees convey the exact same thing: a person attended a dental school in the United States or Canada, completed the required courses and examinations and became a certified practitioner of dental healthcare.
The first dentists hung up their barber’s shears and razors and graduated from the first dental college, The Baltimore College of Dental Surgery founded in 1840, which bestowed upon them a DDS degree. These Doctors of Dental Surgery practiced for a number of years with the only dental degree that existed in the United States. Harvard University decided to get in on the dental school action in 1867, and as they only offer post-graduate degrees in Latin, their graduates were given a degree bestowed with Dentariae Medicinae Doctorae. Today we consider these graduates to be Doctors of Medicine in Dentistry or Doctors of Dental Medicine.
Both of the degrees are subjected to the same requirements overseen by the same accrediting body, the Commission on Dental Accreditation (CODA), to ensure that all dental school graduates are prepared to provide adequate oral healthcare.
All dentists are then subject to continuing education requirements, which vary by state, that ensure contemporary methods and techniques are being learned to provide the best level of care to our patients. Dentists who don’t are subject to license suspension or revocation.
So what’s the difference? Dr. Dow went to the University of Iowa and Dr. Fisher went to Rutgers University. That’s it.
For more on the history of dentistry and other interesting dental things, see the ADA’s website.
