Grilling is cooking something on a grill or open fire, with a direct heat source, with the lid open. Little to no smoke is involved in the process and it’s used for things that cook well over medium to high heats.
Barbecuing is done with the lid closed, over a low, usually indirect heat, often involving smoking as part of the cooking method. It is used for larger cuts and joints that benefit from long cooking times and can handle the flavor of smoke.
Both methods are great, but you need to pick the right one for right dish.
Whether grilling or barbecuing, there are a lot of different methods and you’ll see different methods in countries around the globe. In the Middle East they do a lot of grilled skewered meats, often marinated in lemon juice or yoghurt. In Southern Africa various wild game, kebabs, steaks and sausages are cooked over open flames in a method known as "braai". The Jamaicans have their jerk chicken and fish. In Latin American, the Gauchos were the foundation of Churrasco, a method of cooking meat over open flames. Argentina has asado, or marinade-free meat cooked in a smokeless pit.