Я нравлюсь нечистой силе.
This basically means "The forces of darkness are fond of me," right? (Lit., "I am pleasing to the unclean force")

But Мне нравится нечистая сила is "I am fond of the forces of darkness" (Lit., "The unclean force is pleasing to me")

In other words, krwright, if you keep in mind that the literal underlying meaning of нравиться (+ dative) is "to be pleasing to", it's easier to figure out when to say тебе нравится кто-нибудь ("you like someone") as opposed to ты нравишься кому-нибудь ("someone likes you").

Мне нравится летать не метле.
"I like to fly on a broom"

Десять очков Гриффендору!