I am using some old games like MS HOVER on a 22" 1080p monitor (HP) @ 1920X1080, I switch out the monitor with a 32" SONY BRAVIA TV Monitor set to 1080p (1920X1080) and these old games will no longer display in full screen. Instead they use a half size box in full screen. Why does the OS (XPsp3) care what size the monitor is? It is the same number of pixels, how can it possibly matter how large the screen is? They are both running at 60Hz over a VGA connector.
Note: I see the same behaviour on my brothers 1920X1080 27" Samsung Pivot monitor.
My guess would be that the older game is not actually displaying the game at a resolution of 1920x1080. It's actually displaying at a lower resolution that your monitor can handle, but your HDTV cannot. You should be able to test this by playing the game in a maximized window mode (as opposed to "fullscreen").