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Games for Windows is simply something that gets put on the box art for some games, to suggest that they should work well with Windows Vista or Windows 7. It is separate from the Games for Windows Live (or GFWL) service which can be used to integrate the game with Xbox-style achievements, online multiplayer, and some DRM capabilities.
The retail copies of Call of Duty: World at War is not a GFWL game. You will need to install it with using the CD, and enter the CD Key when appropriate in the installation process.
The application you installed was probably the GFWL Store, which is used for buying DLC for various GFWL games (for example, Microsoft Flight). However, this does not have an option to download GFWL games that you've previously linked to your account through a CD key -- you still need to install them in the way you originally did. And in any case, this is not relevant for this game.
TL;DR: Use the CD to install your game.