"The difference" and "Your thoughts" are two completely separate things. The differences are real and concrete. Your thoughts are just your thoughts.
The fact is, that the difference has always been in the intention. Each discipline has a different mindset. If you intend to train to move quickly and efficiently, that is the mindset of a traceur. If you want to know what freerunning is go to foucan.com.
So what is it when beauty is sought through efficiency? The movements I train are parkour, but since my motivation for learning them is more about fitness and recreation than utility, my practice should probably be considered freerunning.
It's basically another popular band hopping on the parkour bandwagon, but I think they do it a lot better than say, Madonna's epilepsy-inducing video for Jump. God, jump cuts are just the worst thing in the world... The whole point of parkour is flow from one thing to another, and cutting so quickly from scene to scene completely negates that. Awful.
Anyway, in the 3 Doors Down video, a lot of things were clearly done just to make the guy look cool, but look at the flip at 2:25. That's a pretty efficient way to get over a fence where you can't really touch the top, imo.
Parkour = movement to get you quickly and efficiently from point A to point B.
Freerunning = movement to get you quickly and efficiently into bed with a girl you're trying to impress. Or failing that, quickly and efficiently into the hospital.