I just read what the tartans authority has to say on both of them. I understand it now.
The Royal Stewart is a government tartan that people weren't really supposed to wear without permission but it became so popular that they made it universal.
The Black Watch Started out as basically the English equivalent of what the American Company Pinkerton used to be. A Bounty Hunter\Mercenary Company.
Then later on the Black Watch became a well respected army regiment.