How can Colonel Mustang create his flame alchemy differently? When Mustang performs his flame alchemy, he doesn't clap his hands to create it, did he? He's only snapping his fingers:



The others seem to always clap when performing alchemy:



How does Mustang do this?
It's his Ignition Claws. As explained in the 2003 Anime Episode "Fullmetal VS Flame", they create a spark when he snaps his fingers and he uses alchemy to manipulate the other elements to make explosions or fire. The claws have the transmutation circle on them, so if it's removed he can't use it, which is demonstrated during the fight and also later in the 2003 series when he fought Pride.

Ed on the other hand claps his hands because he's been to the Gate of Truth and creates a full array with his own body, thus requiring no circles. It is something only those who have been to the Gate of Truth can do, like Izumi.

While I refer to the 2003 anime, the fundamentals of Mustang's alchemy would be the same as how alchemy works is the same.