I've come across this before, both with Steam's Restore from backup functionality and just simply copying the directory over. a few different methods have solved it for me.
Sometimes, steam only THINKS it needs to download stuff. Clicking play, it seems to figure out that it has everything it needs, stops downloading and launches the game.
Verifying Local Game Cache
Right-Click the Game->Properties and select the
Local Files tab. Click
Verify Integrity of Game Cache
Steam will run through and verify that all the files have been downloaded and are ready to play. If any files are missing it will download them, but it wont be x/12Gb, it should only download missing files.
Delete and Re-restore
Right-Click the game->Delete Local Content will delete the local files (obviously) but it will also stop any download in progress.
Re-run the restore from Steam Backup
Delete and Copy Directory
Delete Local Content, except this time when it is done, close steam (completely, make sure it isn't running in the taskbar), and copy over the TF2 directory from one PC to another. Restart Steam, and click Play. It should just start, although it might run a
Verify Game Cache by itself.