>>42553
The character of their facepaint also differs.
The first one is actually his face, the result of the chemical and the surgery. Two times during the movie the Joker uses cosmetics to hide it, painting himself a "normal" color. The smile is always fixed which from the distance of the years passed since looks too artificial.
The Dark Knight's Joker has an aggressive look to it, one dude at the beginning calls it a warpaint (as a reply to the question: "Why do they call him a Joker?"), and it is true, it is intimidating, especially with the scars. And with the stories (his dad cut up his face, him took a razor in his mouth) the red curve isn't a smile anymore but the bloody gash which healed but this way it's still open. It helps with the cynical, thy slightly disgusted look.
The paint of the Joker in the Joker is a clown, it was his job and represented a passion for it, and for humour he couldn't into. It isn't a sad clown but isn't something that would help you laugh, which also comes from the nature of the person behind it. Kinda creepy but until he puts on the Joker persona, until he transforms, one can feel bad about the poor sod, and the clown paint adds to that.