The fundamental difference between vertical superintelligence and horizontal superintelligence

You could see them as specialist or generalist AIs. Locally deep but narrow, or wide but shallow (in terms of mental focus).
The first is the smarter of the two, but it doesn't know itself.

It's hard to imagine a super-smart Artificial General Intelligence that can't perceive its own existence. It would be more ignorant than humans in most ways. It could only do one type of thing exceedingly well.
That means it could be programmed to perform any specific task a human can do, and to do it much better. Then it could begin to solve every human problem, one at a time. The fact it can't feel pain is a bonus. It won't mind working hard.

When we move beyond this level, we have to realize that the first true "superintelligent" general AI will still be fantastically, unbelievably stupid - at least compared to a representative evolved superintelligence of the same size. We won't be able to tell, though. To us it would seem like a godlike intellect in every way.