DMD is a Doctor of Dental Medicine. DDS is a Doctor of Dental Surgery. There is no practical difference between a DDS and a DMD. The curriculum for each is identical.
Each dental school chooses which degree to award its graduates. As long as the school is accredited, the graduate is eligible to sit for the national board as well as any regional board or state licensing board.
Any licensed dentist is allowed to perform any dental procedure provided they have not limited their practice to one of the dental specialties (oral surgery, orthodontics, endodontics, etc.).