The traditional design of a SxS "game gun" is optimized for an instinctive style of shooting, what many call the Churchill method. The straight hand stock, splinter fore end and double triggers are made for quick shooting. The weight and balance is designed to accommodate a "snap shot". Of course, there are exceptions, and the American SxS's are largely duck guns with more weight, beavertail fore ends and pistol grips.
The instinctive style of shooting demands proper gun fit and a practiced, consistent gun mount. It is very difficult to shoot an English pattern SxS well if the stock of the gun is not well suited to the shooter. An U/O is more forgiving in this respect.
U/O guns are for the most part single trigger, pistol grip designs better suited to a swing through or sustained lead style of shooting. Usually a little heavier than an English pattern SxS, the weight supports the style of shooting they are designed for. Many people claim that the single barrel sighting plane offered by the U/O is easier to use, and that may be true for someone coming from a pump or auto, but I believe that you should be looking at the target, not the gun. I am not really conscious of the "sight plane" while shooting.
Because it is optimized for the deliberate style of shooting, the U/O dominates the clay target sports where the origin of the target and its approximate flight path is known. Grouse or pheasants often demand an instinctive shot, you don't know when, where or in what direction they are going to fly, and seldom offer more than a brief moment in which to make the shot. This makes the SxS a better tool for that use. While some U/O's have the weight and balance to accommodate an instinctive shot, none of them rest so low in the hands and provide so quick a mount as a well made SxS.
Yes, the English have made heavy fowler SxS's, and yes, some double triggered straight stocked U/O's exist, but I do not see the stack barreled guns as competitors to the SxS's. Rather, they are two distinct designs meant to fill two distinct niches in the shooting world. Target shooters in particular have a way of denigrating the SxS as an antiquated design, but I have yet to find a better tool for instinctive shooting. The existence of modern mass produced SxS game guns such as the Ruger Gold Label shows that I am not alone in my viewpoint (even if the Ruger has only one trigger - a concession to modern American tastes).
Like an F1 race car compared to a NASCAR stocker, both are highly tuned for their specific tasks. You can drive a Nextel cup car on a road course, and you can drive an F1 car on an oval, but neither is very well suited to the task. How finely the gun is tuned for its purpose in life is one of the measures of how well the gun is built. You are not likely to see a SxS win the world trap championship, but by the same token a man carrying an 8 1/2 lb. Perazzi through my sharptail fields is at a distinct disadvantage.
This is only my opinion, and many may disagree (feel free). My opinion comes from experience with both types of guns in both clay and upland game shooting. Try it for yourself, and see what conclusions you come to.
Sharptail
