How does food cook differently on an open top grill than a covered one?
I believe it is kind of like the difference between broiling and baking. "Open top grill" only cooks from underneath (kind of like broiling but not from above). Baked or "covered" grill will distribute the heat more evenly.

For more information see: http://www.epicurious.com/articlesguides/howtocook/primers/grilling_lightingandgrilling

That's where I double checked my assumption and got the metaphor of broiling and baking.