We really get to know their personalities and they get to be a real companion. They make us laugh, they make us happy, they make us frustrated, they make us cry, they make us sore, but they also heal us. They may only look like livestock, but they're individuals; no two are the same.
From a sporting standpoint, It's almost like a puzzle a rider has to solve; they have to know which buttons to push and which ones to stay away from, which makes horseback riding the hardest sport AND the most addicting sport, in my opinion.