If he hadn't been sidelined with a 14-month neck injury (February 2003 to April 2004), then as already touched upon by JC, he probably would have become a world champion a lot earlier than we did. In fact, Wrestlemania XX would have likely been the culmination of Edge's climb to the top instead of Eddie Guerrero's. (This is debatable and not at all set in stone, of course, but it's highly likely.)
And if he hadn't of been forced to retire in 2011, we would have witnessed some great rivalries like Edge (and partners) vs The Shield, Edge vs Seth Rollins (I sense that he and Rollins would have been some great chemistry with one another, probably in large part because Seth's heel persona at the moment is similar to Edge's from 2005/2006, and they could have feuded over the MITB briefcase), Edge vs Randy Orton (he would have made a good opponent for Orton during his god awful championship reign from late 2013 to early 2014... these two have history, both as allies and as adversaries, and anything would have been better than seeing Big Show getting another title shot), Edge vs Brock Lesnar, etc.
All in all, he probably would have already begun his descent into the Mick Foley/Chris Jericho role of jobbing to practically everyone he wrestles. Not that that would necessarily be a bad thing at this point in his career.