From Merriam-Webster, grilling cooks with heat from above, or on a flat surface like a griddle (grilled cheese sandwich, yum!). Barbecuing is usually over hot coals, but can also be rotisseried over heat.
It may mean different things in different countries, I'm not sure. In Australia, a bbq is cooked outside either on a hot plate or on a grate over coals. Even if you cook it the same way inside the house it is not called a bbq. If we grill meat, it is cooked under heat. Mmmm, I would'nt mind a grilled lamb chop now, with mashed spuds.
Often the words are used interchangeably. However, in the U.S. "barbecue" can mean smoking the meat and basting it with sauce. Different regions in the U.S. have different styles of barbecue. Actually you may need a cooking website, not a language website, to fully answer this one.