Raptr is some odd social networking thing. Probably useless to most people.
"AMD Gaming Evolved" is a client that connects to Raptr. Probably also useless to most people.
Crimson is the name they attach to their driver package and related software. Previously, it was called Catalyst. (I don't know where they come up with these names. I nominate "Exothermal" for their next rebranding).
My question then concerning this software is--is the Crimson an updated version that enhances the overall GPU performance of the computer, or is it meant strictly for gaming performance?
It would represent a new driver version, I'd expect. The specific changes could be only related to games, or it could resolve major issues with desktop application functionality. Keeping the existing version (assuming it is working fine) won't hurt anything though.
To get rid of the prompts you should be able to open Task Manager and go to the Startup Tab, and right-click and "Disable" any items related to AMD.
You might be able to find the Raptr and "AMD Gaming Software" items in the control panel's Add/Remove programs and remove them.