They have a difference, Horses with the "Horse Type" tag are horses to be ridden. And then they state if they are a "regular" horse or a war horse. The "Type" Tag is what type of item it is.
So far that's what I've figured out. The sumpter Horse is a pack animal, even if you can ride it, and cannot be used to upgrade troops. It also does not have the "Horse type" tag.
In the end, it's for differentiating if the horse is a pack animal (No tag), a normal riding horse for low-level cav (Horse), and a high-level horse for elite troop upgrades (Warhorse).