Can I use this to control a robot over the Internet?
What about this one
Absolutely.
What's the difference between those two?
I have an arduino mega 2560 is it compatible?
Do yourself a favor, and don't get the gainspan module. I salvaged one out of an RC car, you have to sign an NDA to do anything. Hours and hours of work and I've just gotten it to show up as a WiFi AP.
Get the wifly module.