The most obvious is that he is not the World Chess Champion, but world #1 - the difference being that you have to defeat the previous champion in an extended match in order to claim the title.
Second, both Carlsen and Kasparov think that Carlsen is stronger and weaker than Kasparov in certain areas. What they have mentioned to the public before is that Carlsen has a better intuitive sense of a position, while Kasparov has greater facilities of calculation. What they both agree on, it seems, is that Carlsen, in large part due to the aid of computers, has reached such a high level of ascendancy in chess at a significantly younger age than Kasparov was able.
Finally, as he plainly states in the article, Carlsen is trying not to be crazy, by living as normal a life as he can and not becoming obsessed with the game to an unhealthy degree.
Aside in relation to a response to your post, since I don't feel like posting twice: It is insanely difficult, if not impossible, to say that Carlsen would beat Kasparov - especially now that Carlsen has been using Kasparov's advise on openings against other opponents, we observers cannot tell what recent innovations have, in fact, been his original work and not that of Kasparov. Another point to consider is that, even with the amount of rating inflation there has been over the past several years, Carlsen's 2813 rating does not come close to Kasparov's maximum of 2851. Finally, Kasparov was uncontested at the top for nearly two decades, whereas Carlsen is still having troubles overcoming some other top grandmasters (specifically Kramnik of late). Unilaterally saying that Carlsen would be Kasparov in a match is, realistically, a rather uninformed opinion.