They mean the same thing. Anyone who thinks different is WRONG!
First, nobody really even knows how to spell one of the terms, so it isn't that far fetched that nobody even knows what they are talking about when using a word nobody even know how to spell. Half the time, it is spelled BBQ and half the time it is spelled barbeque, while it is spelled barbecue the remaining half o the time.
Second, when used as a noun for an object that you cook on outside, many even use both names to refer to it, as in bbq grill, barbecue grill, barbeque grill (perhaps even Barbie Q. Grill if you are really into Mattel dolls or got some hot blond girl cooking outside for you). If it means the same damn thing when used as a noun, why would they mean two different things as a verb?
Third, there is a noun for an object that is designed to cook "low & slow" outside. It is called a smoker, even when you don't use it to smoke. If barbeku meant slow cooking, a smoker would just be called a babreekew and everything else would be a grill.
Fourth, when used as a noun for an event, barrbicue is the term that is always used, even if one is cooking fast & hot. NOBODY ever invited ANYBODY over for a "grill", but they might invite you to a barbecue where they are going to "grill some steaks" and have some beers.
Bleeding over from another thread, I wonder what Hal would say?