Amazed is much more intense than surprised. I was surprised to learn that she was a math student. I was amazed to learn that the guy next door was a world champion athlete in track. I was surprised by her call. I wasn't expecting it. I was amazed to see that everyone walked away from that horrible accident and that nobody was injured.

Amazed means an overwhelming feeling of wonder and can last quite a while. In comparison, surprised means something sudden and new which takes over your attention at that moment. It's quite quick, and doesn't last so long.

To expound a bit upon Maxey's excellent response, I would say that "amaze" usually has a positive connotation, whereas a "surprise" can be good or bad.