Sorry, but I don't know exactly what the driver is used for by ATI. I can't even test any of this as all of my systems are Intel CPU + NVIDIA video. The reason: ATI video shows up in BSODs 1000 times more than NVIDIA.
i.e., I don't know what to tell you on how to update that driver as I have no experience with it whatsoever.
Try uninstalling CCC and installing the lone ATI video driver.
Make sure that the HDD that you installed Windows on while it resided in that system is back in that system. Windows doesn't do well with "foreign" (switched) drives. Doing so can result in Windows declaring the OS to be non-genuine followed by going into reduced functionality mode --> https://support.microsoft.com/en-us/kb/925582
Regards. . .
jcgriff2