There are some big differences between those riding styles for both horse and a rider. Let's start with the differences for a horse.
"English" claim that a horse has two ends, front and rear. Their idea is to propel it's rear end by pushing it with calves and to brake it's front end by pulling the reins. They call it a collected horse. Similarly to pressing a spring, which can be extended. While on trail, the reins are often used like a support for rider. It's hard to figure out when a rider is giving a command and when he is normally pulling the reins and when he has lost his balance and is just fighting to stay in the saddle. It's a sad truth that there are few riders who don't hang on to the reins. Sad for horses.
"Cowboys", who have been in the saddle all day long chasing cattle, have developed a different riding style. As long as the horse does what it's supposed to, it is rewarded by absence of any coercion. The reins are loose, forming a semicircle, and the legs, calves aren't squeezing the horse. if we want a change in direction of movement, we shortly make a contact with the bit until tho horse changes the direction, and as soon as it obeys, it gets rewarded by loosening the reins. It's easy to understand for a horse because nobody likes to be poked in the mouthe with a metal bit.
For riding in a sandy round pen and show jumping, english style is obviously good. Would a western style be better for show jumping is something we'll never know for a simple reason that show jumping is allowed only to english riders. Western style is not allowed. Things change when it comes to riding in a difficult trail. For galloping through forest, it's enough about 30western-style riding lessons, and for the same thing it takes even years for english riders. A different leg position makes it harder to ride english style on trail where horse normally changes direction and speed on it's own, not to mention sudden stops. It often results in a flight over a horse's head for english riders.
There are few differences in regards to the rider, but two are most important.
Feet position in english riding style is parallel to horse's spine. It's impossible to lower the heels in that position because it's not an anatomic position, and nakles hurt in that position. In western style, feet are positioned under 45 degree angle in relation to the horse's spine. It's a natural position which enables you to lower your heels as low as possible in order to keep you from falling off, especially to the front.
A neutral body position often gets mentioned in literature. While neck reining, we must simultaneously use the reins, our legs and body. And we never get to "hang" on the reins, so neutral body position is easily and quickly learned.