Most things on the xbox (or PS3 for that matter) don't render at anything higher than 720p. If your console is set to 1080p, it just video scales the render to whatever your settings are at. (There are games on both systems that render at 1080p, it's just that the majority are at 720p)

But if you set your TV below 720p, the game might render some objects (like text, or HUD graphics) a little larger in comparison to the rest of the scene... so you might find that works better on your 19" monitor... unless you're just sitting 3 or 4 feet away from the monitor.