The difference between western riding and the riding that's traditional and established throughout most of Europe, called English by western riders, isn't all that significant.
Western riding can be roughly divided into two categories, those who just want to trail ride and have a good time, and those who want to show and have a good time. Those who want to trail ride western should be just as concerned about doing it right as those have to be who present themselves to the scrutinizing eyes of horse show judges, because they owe it to their horses.
The most obvious difference between English and western is that the reins are held in one hand only, at least with mature horses. Horse show rules offer classes for junior horses in which they may be ridden with two hands, but the finished western horse is ridden with just one hand on the reins. Show rules also request the reins to be held in such a way that not more than one finger is between the reins. Good western horses are also ridden with slack in the reins - you just can't impress a judge riding a tight rein. This calls for a training program designed to work more on your horse's mind than on his mouth.
Western horsemanship denotes a style of riding that is something of a departure from the traditional English riding style. Instead of a focus on highly stylized forms, the western technique relies more on the horse's natural gait and inclinations, leading to a more casual, fluid style. Not surprisingly, this style is better suited to the long, rigorous trail rides that are common out West.
Additionally, the western style is built around leading a horse based on what the horse's instinctual tendency tells him or her to do, while still maintaining full control of the horse. Also, no force is used to control the horse; instead, the horse is simply guided in what he or she needs to do. This style is highly effective for maintaining optimal performance throughout a long trail ride.
Leg cues are largely the same as in English, but a good rider/horse team needs less of it. Normally, the western rider doesn't use the legs just to maintain a gait. Weight cues are applied differently by most top western trainers, though. The western dressage horse is taught to move away from pressure, including weight. This means that the rider's weight is rather on the outside of the horse in a turn-around or a circle, pushing the horse in. If it's a well-tuned team, the rider would be sitting pretty much in the middle, but the principle of steering is one of pushing/driving the horse into the direction it is supposed to go, in contrast to putting your weight there and expect the horse to follow.