Generally you should cook with your grill lid closed. The exception to this would be for those cases where you want a quick searing but not a thorough cooking.
Closing your grill’s lid allows the heat that has risen up from your heat source below to be reflected back throughout the entire grill. Doing this enables your meal to be cooked not just from below but from all sides. This is essentially the same reason why you would want to close your kitchen oven’s door.
Closing the system circulates the hot air like a convection oven. It also obviously is more fuel-efficient and faster.
From a flavor standpoint, when barbecuing with the lid closed, your food retains its natural juices and moisture. It also allows those vaporized drippings to be better absorbed, creating that awesome backyard barbecue taste.