There is so much to learn about Western saddles. Not only the types mentioned about, but the rigging. They have many different types of rigging; 7/8, 5/6, centerfire, in-skirt, in-tree.
Western saddles really should ALL be rigged with two cinches, a fore and a back. But many people neglect the back cinch, including me. However, the saddle will not fit as well as it should unless fully rigged.
Western saddles have a larger tree in them, thus they spread the weight of the rider out over a larger surface area, so make it easier for the horse to bear the weight. However, many modern western saddles have trees that are what is called "downward sloping", such that they kind of end up putting too much pressure on the front half of the saddle, especially when not double rigged wiht the back cinch.
I haven't ridden in a cutting or reinig saddle, only a trail and a barrel. I don't like the way the barrel locks me in, nor the high horn. The trails I think have the stirrup bar a bit more under the thigh to allow for a more straight downward drop of the leg, less of the chair seat that you would find usefull for roping and maybe barrels, too. there is a place where a chair seat is helpful.