As I roughly understand things the general train of thought for the D-Day landings in Normandy was successfully land, get far enough inland to build up a decent defensive perimeter, join the different landings together, build up your forces, capture a port like Cherbourg to secure a supply line and then break out. The Commonwealth forces were to attack Caen and pin down the German troops and hopefully provide and easier breakout for the western troops with the American forces then following a modified plan to swing around on the Commonwealth pivot/hinge. One of the drawbacks of this though was that it caused the Commonwealth forces quite a large number of casualties, something they couldn't really afford - IIRC it was about then or a little later that the British started having manpower problems, whilst, to be blunt, the US never really had that sort of worry with a ready supply of bodies to chuck at things.
So what might happen if the beaches and roles had been reversed? US troops land on what in our timeline were Sword, Juno (and possibly) Gold beaches with the Commonwealth forces landing on Omaha and Utah. Other than reduced Commonwealth casualties would it make much of a difference? Only thing I can think of it possibly a slightly more successful closing of the Falaise pocket but that could really go either way.