The movie starts off with the Joker straight-up murdering Jason Todd, the second Robin, who is later revived by Ra's al Ghul. Driven by revenge and anger, Jason then tries to become the new crime boss of Gotham, which finally culminates in a stand-off between him and Batsy:
During the confrontation, Jason asks Batman why he didn't execute the Joker after the clown killed him, to which Batman replies, "All I've ever wanted to do is kill him. A day doesn't go by I don't think about subjecting him to every horrendous torture he's dealt out to others, and then end him." The idea of Batman, a man with a genius IQ and Olympic-level physique, fantasizing about torturing someone to death is one of scariest things ever.
Burton's Batman was definitely creepy and all, but he was ultimately an unhinged psycho. Under the Red Hood, though, makes Batman look human, and then twists that humanity into something truly dark. There's even an earlier scene when the Joker talks about how Batman found him after Jason's murder, and put him in a body cast for six months. I've envisioned this scene many times and still can't decide which version is scarier: Batman going all berserk on the Joker, or calmly and methodically breaking every bone in his body.
Although, in my imagination, the Joker doesn't have Jay Leno's chin.