Gunfighter shoots both revolvers at the same time, usually, but not mandated, alternating guns between shots. Double Duelist, by definition, is one revolver at a time, but one shot lefthanded and one shot righthanded. Some stages are not set up to legally be shot in the Gunfighter style (eg 5 pistol shots followed by a rifle or shotgun sequence, as a Gunfighter may not by rule, reholster handguns with unfired rounds) in which case the Gunfighter would shoot Double Duelist. Double Duelist is not a separate category, however a competitor registered as a Gunfighter could concievably shoot a whole match Double Duelist and be within the current rules.
Hope this helps.