In American terms, I agree that to grill something and to barbecue something are technically not the same thing. I think both are understood to mean cooking with open flame (traditionally wood or charcoal, although it has expanded to include gas). Grilling, as has been pointed out by others, usually involves cooking quickly over direct heat, whereas barbecue usually means cooking for a longer period over indirect heat. You would generally grill a thin steak or a hamburger, but you would barbecue a whole pig or a leg of lamb. (As always, there are exceptions.) You can use either of these methods on something like a Weber kettle. If you cook a steak directly over charcoal, you are grilling it. If you pile the coals in a corner and put a rack of ribs on the other side for hours, you are barbecuing.
The differences are real for people who care, but I think that is a small group of people. For most people, the terms grilling and barbecuing have become more or less synonymous, and the item you cook it on is interchangeably called a grill or a barbecue (or a barbecue grill). To correct someone would be pretty pedantic (this is the nicest word I could come up with). If my neighbor invites me over and says he's going to barbecue hamburgers and hot dogs, I am going to say "thank you" not "you're grilling, not barbecuing." If he says he's had ribs on the grill for hours, I will ask if I can bring a side dish not "you know you're barbecuing, right?"
I don't think any of these terms necessarily apply outside of the United States.