Typically, grilling is fast cooking, things like burgers, dogs, etc. even steaks. BBQ is considered low and slow cooking, brisket, pork butts/shoulders, bigger pieces of meat.
To me it really doesn't matter much with all the technicalities. Whatever it is tastes better if it's done on the grill, smoker, etc.
For tailgates, you really don't have time to low and slow it or smoke anything. I typically fire up my grill about 8:45 and we're walking to the stadium around 11:20. 20 minutes or so for the charcoal to ash over, and that leaves you about 2 hours to cook, eat, and clean up before you go to the game. Actual cooking time is no longer than about an hour and a half or so at the max. That really doesn't give you enough time to "BBQ" or smoke.
That's the best picture I could find of the grill I use. And I've cooked or "grilled" everything from burgers to pizza to meatloaf at tailgates on it.
BTW, I think that's Philly cheesesteak on the grill in that picture.