What we now call "western" riding was invented by the Spanish for the purpose of - chasing cows. The western saddle is more similar to a lancer's or jouster's saddle than the lighter saddles used in other modern styles, with a higher pommel and cantle designed for a more secure seat. (Australian seat is very similar). The western saddle also has an elevated horn - useful for hanging things off of. Including yourself in a tricky situation.
The biggest difference between the western style and other styles, however, is that the western rider typically keeps only one hand on the rein. In other styles, both hands are kept on the reins outside of emergency situations. The horse is trained to steer from a touch of the rein to the side of the neck and the bit is used only to stop (and then only when absolutely necessary). Because of this, western riders use apparently harsher bits than English. The reason? The cowboy needs to keep a hand free at almost all times, be it to use a rope, open a gate, drink a canteen, etc. In America, the western style is preferred for long distance trail riding (outside of endurance racing, where the 40-50 pound saddle is considered too much weight) and especially sight seeing. You can use a DSLR from a western trained horse. Not happening when riding English.
For worldbuilding, something like the western style could well be developed by any herding culture. In the modern world, western is more popular in most of the United States and Canada and, as you might suspect from the name, becomes more popular as you travel west. It's rarely seen in Europe outside some parts of Spain and southern France (and when it is, it's only seen in the show ring as an interesting variant). It's also more popular in Australia and New Zealand.