The title pretty much says it all, 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.
It's the difference between radiant heat, and convective heat.
Radiant heat is great for searing, and is produced by your charcoal/burners. You're always going to have (approximately) the same amount of radiant heat, regardless of whether the top is up or down.
Convective heat is just as good for cooking, but you're not going to get the delicious crust. When you have the grill covered, you're going to get a lot more convective heat, and uncovered, a lot less.
I won't go into the physics of it, because it's not terribly interesting. Basically, on an open grill, you're heating up one side of whatever it is you're cooking, and letting heat escape from the other side. The flame itself is also hotter, because of the better airflow.
When you close the grill, the fire itself is cooler (still enough to burn you though), but the heat remains locked in the grill. That way you get a more even cooking, and it's actually a bit faster, because the environment of the food is much hotter than it would be in open air.
There is also a secondary effect, where closing the grill traps some of the smoke in, and gives a smoky flavour to the food.