Have you attempted to install these games on both systems?
If not, then we can certainly rule out the evil copy protection.
Another thought (if its is only the game discs): Perhaps these games are not compatible with 64bit Windows (and the amount of RAM you have installed).
Perhaps you can do some more experiments to find out what works, and what does not work in order to narrow down the problem.
Are you able to copy the content of such a DVD to a folder on the HDD?
Another possible source of trouble could be a compatibility issue with the sata controller that is used. Make sure you have the latest drivers for your storage controllers installed. My personal preference would be to have the Sata ports used by the optical drives configured as “IDE” and using MS default drivers. But this is often not an option “for everyone”.
But you could try another Sata cable and switching the drive to another Sata port (perhaps there is even a remark in the motherboard manual (like “Sata 5 and 6 are for ODD”) about that).