I've been taking this from a geologic standpoint because that is where my background is. For that, plates works well. But obviously, on a bigger, more holistic level, 'continent' becomes more vague. I think talking geographically and culturally, it's a historical thing largely. As Europeans developed the concept, they created these separate areas because (1) they represented a significant travel time to reach and (2) they were generally culturally different enough to be seen as distinct. Old (again, European) maps have the center in the Mediterranean, not the Atlantic. It was easy to seperate the world into Europe North of the sea, Africa South, and Asia East.
Now, I think a lot of the hold over is for ease of education. It's much easier to teach kids based around readily divisible sections with obvious divisions. Africa is an easier concept to describe than say "Middle East." (Which really isn't any better defined than continent.) Also, helps with teaching geologic history, and cultural history. Other than that though, I too see little value in it, especially in a globalised world.