True, he is a freak, and, true, the makeup clearly can work. But, don't you think that permawhite would only serve to cement the fact that he is a true freak?
With the makeup, he's a freak on the inside, sure, but, on the outside, he's not that different from you or me. It's only half there. So, he must add the clown imagery himself. But, just as he puts it on, it could be taken away. In removing that, you've removed part of him.
Permawhite makes that impossible. It can't be removed. He's all Joker, all the time. Inside and out. Without his makeup, it would be the equivalent of Batman being's identity being revealed to the criminals of Gotham. Sure, he's still capable of kicking their asses. But the effect of the bat is gone.
And, this is just personal opinion, but I find permawhite to fit the Joker better as a deformity than the bubbly mouth scars. It's something really out there, to the point where it's almost unbelievable. It defies logic, just like the Joker should.
It's also visually jarring. The chalk white flesh all over the body, it's just very bizarre to look at.
Plus, symbolically, it represents everything that the Joker is: on one hand, it gives him the appearance of a clown, and, on the other, the visage of a corpse or a ghost. It represents things both innocent, and at the same time, morbid...just like the Joker.