Why is YF22 designed with conventional rudders while YF23 with all-moving rudders?
As a general rule when you see an all-moving surface vs a fixed fin and control surface it's because you get more control authority per unit of surface area, so the surface itself can be smaller.  Better stealth, less drag.  You lose the passive weathervaning effect that provides basic yaw stability you get with a fixed fin, but with a fully active artificial stability system, that might be acceptable to the designer (although maybe not to the Air Force).

The 23 went with what amounts to a V tail, on top of all-moving surfaces, with the surfaces providing yaw and pitch authority at same time, and a fully active artificial stability system to make up for the lack of fixed surfaces.  This would have had much less drag than the regular horizontal and vertical surfaces, and probably a better radar signature to boot.  

Why it lost out to the 22? According to Wikipedia, it was faster and had better stealth, as your would expect from the tail arrangement, but was less agile, so we know what the Air Force's priorities were. And there was probably a range of other factors that made the 22 a better choice overall (it could be that the 23 was just a bit too radical, which may have impacted maintainability, reliability, and cost factors in a way that made the 22 look like a safer bet).