The rivalry itself is just like any other rivalry. But I believe the lore behind it has embedded itself into American culture like few others have--I think it has become synonymous with rivalries. You ask a casual fan the biggest rivalry in college football? They'll say Michigan/OSU. Basketball? They'll say Duke/UNC. Baseball? Red Sox/Yankees. It's not that the games are always greater or the teams are always better, it's simply that it has reached a point where when people think "rivalry," they think of those I listed. I think of it like the fact that Babe Ruth is the greatest baseball player to ever play. It's not true (the answer is Barry Bonds, btw) but it has become a thing of legend and a general consensus to the point where it is simply considered fact.
So when I say the SEC doesn't have anything that really comes close, I mean it in that sense. Of course Bama/Auburn or UT/Georgia or LSU/Florida are equally heated and a great rivalry. They just don't garner the same national intrigue of a Michigan/OSU.