"Spite" has to do with causing regret to another. "To spite someone" means to do something to cause them to regret.
"In spite of" is used without the harsh meaning, and means something happened or was done sucessfully, regardless of (or even opposed to) the situation. Robert's explanation of "even though..." is pretty accurate. You're not actually causing regret to someone or something.
The situation is not always negative, just opposite to what finally happened: "In spite of having lots of money, she didn't buy the dress."