You can have horses in a pasture OR grow hay. You'll need someplace to move them if you want to grow hay in the same pasture. Plus pastured horses eat round bales, so you'll have to have those as well as the square bales.
I'm in the process of getting all my geldings into pastures so I don't have such a big hay bill every week. And also having enough room I may stables so when they get barren I can move them into the barns and let my Feilds grow back and then move them back and repeat lol
They are basically the same besides you can grow hay in a pasture and if you have unbred mares and stallions in a pasture they have a chance of randomly breeding