Charles usually wears a tartan linked with one of his Scottish titles, Lord of the Isles Hunting or Rothsay Hunting.
Her Maj often wears Stewart which Queen Victoria was entitled to through the female line, and there are several versions of it for different occasions, Old, Royal, Dress, Hunting etc.
Only HM can wear Balmoral tartan unless she invites you to! It was of course designed by Prince Albert! 
Like so much of the RF tradition, {being focused on charity work for instance}, wearing tartan is quite recent.