In all ovens, and covered barbecues, and grills for that matter, a certain amount of convection is working. From a scientific standpoint, convection is one of the means by which heat is transfered into something, and is a natural process that seeks to bring everything to an equal temperature. That is, all the surfaces closest to the heat source rise in temprature, usually due to radiant energy absorption. These surfaces contact the air and transfer some of the heat energy by simple conduction. The warm air then rises and displaces the heavier "cooler" air which sinks to the oven bottom, to be heated and itself rise upward. And the cycle continues. The top air cools as it gives its nergy to the oven walls, the food, etc, and is replaced by the rising hot air. Thus, the oven get hot.
Now a conventional oven has a fairly small area and little convection occurs. The oven stays at a fairly constant temperature from top to bottom. But there is a difference. There is a reason you don't bake the cookies on the bottom rack.
Conventional ovens transmit their heat to foods through convection - the continuous movelment of hot air against a surface, conduction - the heat is conducted through the metal oven walls, and is transmitted through the metal to the racks, to the pan, and to the food through touch, and by radiation - Heat is radiated from all hot surfaces and is absorbed directly by the food and pan.
To understand a convection oven, think of a warm day, with a cool breeze. The radiation from the sun is absorbed into your body, raising your body temperature. In still air, the only cooling mechanism you have is evaporation. You feel hot, uncomfortable. The still air is only able to absorb so much of your excess heat, and the moisture evaporates slowly. But then, the air moves. Now it's the same temperature as was the still air. But since it's moving, new air replaces that which was warmed by contact with your skin. It also causes faster evaporation of your sweat. The new air is cooler, relative to your skin than is the air that was warmed by your skin. Therefore, there is a more rapid exchange of energy. your body is cooled more efficiently. You feel better.
The same thing happens in a convection oven, only in reverse. The air is hotter than is the food. And the moving air is able to both absorb more energy from the heat source, and transfer that energy to the food due to fresh hot air always touching the food surface. Also, the moving air helps elliminate hot spots cause by radient heat, or still, slow-moving air. The result is faster cooking with more even heat surounding the food.
The advantages are obvious. But the disadvantages exist as well. Radient heat energy is somewhat neutralized as the moving air takes away the intence heat created by thermal radiation on the food surface. Foods just don't brown as well. The air must do the lion's share of the work.
As stated above, both ovens have their place. Which one is right for you depends on what you most do with your oven.
And the covered grill, it cooks mostly by radiation, even with the lid closed, as does your oven's broiler. Moving air has little effect with these appliances.
Seeeeeya; Goodweed of the North