So i like them, and they do have a natural chemistry, and they are nice guys.
As far as what's "real" though, they are scripted and rehearsed down to the tiniest detail - and they don't write their own lines. All the "natural banter" you see is for the most part given to them (originally actually at times by David Walliams and a few others). This goes right the way across all their shows.
They HATE their past. So much so, they bought up the rights to their music and old shows (especially Byker Grove) so they could control when it was shown. What is happening with LGRTR is interesting because they deliberately tried to take their songs out of the public domain. Very occasionally they have gone along with doing one-off's a few times - like this - so either they are mellowing or they went along with it because they knew it would be turned into a viral thing and promote the show which is obviously suddenly getting a massive ratings boost they want to keep up.
The public do love them, and they are put across well on TV. I think videos going "viral" are just as much about the marketing of it to the public - if theres loads of things in the press saying "get them to number one, its amazing - the best thing ever" - to a degree, when its a name like Ant and Dec they will go and have a look.