Answer:
The degrees DDS and DMD are the same and earned by by attending an accredited Dental school. Initially Dental schools in the United States were not associated with any university setting, rather were separate colleges scattered throughout the US. The original degree the awarded was DDS (Doctor of Dental Surgery). Harvard university later decided to form a Dental school. However Harvard only awards degrees in latin, thus they decided to change the degree to DMD - Dentariae Medicinae Doctor (Doctor of Dental Medicine). Throughout the years the smaller separate schools scattered throughout the US basically merged with the States Universities and continued to award the DDS degree. If a University system starts a Dental School they can decide to award the DDS or DMD degree, however most elect to award the DMD degree.
In regards to training, DMD and DDS are the same, all licensed dentist in a given state must pass the same test and have the same schooling to practice dentistry.