The obvious advantage of a gizmo like this is that it lets you climb on your own without a belayer. Also, you can train continuously without having a belay slave. The major practical disadvantage is that it doesn't lock, so there's no way to hang and work out a move. If you fall, you have to start over again from the bottom, which means that it's a lot harder to work through difficult moves, since you're tired by the time you get there.
Psychologically, though, it's even weirder. When someone is belaying you and you fall, the rope stretches a bit (assuming it's a dynamic rope) but then you stop dead. With an auto-belay, you just fall slowly. If you're used to a regular belay, your first thought is "my belayer has screwed up and I'm about to fall to my death". That's not really an easy reaction to suppress, which makes it a lot harder to climb near your limit, as well as making letting go to descend at the top of the climb a real act of will.