Well if you want a somewhat rational explanation Harvey Dent could have been suffering from Vitiligo. As he started to slowly lose his skin pigmentation he covered small light spots with dark makeup to match his natural complexion. But as it progressed the majority of his face was without his natural color so he started to cover the dark spots. So he did not need his whole face covered with makeup. This is how people with the condition sometimes deal with it.
Imagine the added effect of having acid thrown onto half of your face if you already had Vitiligo. He had lost most of his natural complexion and now he had permanent scars too. Talk about the loss of identity and psychological damage!
I know none of this is in the movies, buts its an interesting way of tying two actors playing the same role in the same continuity together.