Response time is a big one for me. I can perfectly compose a shot with my point-and-shoot Panasonic DMC-ZS3 and lose it because of lag-time. When shooting wildlife, pets, active children, etc, you can miss the "perfect" shots, especially when shooting in automatic modes in which you're relying on the camera to make decisions regarding shutter-speed, ISO, auto-focus, etc.
I have two Canon 30Ds with a modest array of lenses. Those cameras can blaze away at 5 frames per second, and they have helped me catch some things I would have missed otherwise, like a really precious expression on my niece's face before she knew that I was shooting her. (She was already camera-conscious by the age of 2!)