There are three things about this movie that strike me as unique.
The first is that the movie isn't about Totoro. The movie is about the two girls, Satsuki and Mei, and revolves around them moving into a new house and going to a new school and being deathly afraid that their mother is going to die. Totoro just shows up every now and then.
The second thing is that the fanciful characters in this movie are not fully anthropomorphized. Both Totoro and the Catbus have huge mouths filled with teeth, and look as if they could bite the girls' heads off at a moment's notice. And so we are forced to judge them by what they actually do, instead of (in the Disney manner) what they look like.
The third thing is how understanding all the grownups seem. The clearest example of this is when the father and Satsuki find Mei, asleep near the base of the great tree. When Mei awakens and tells them about Totoro, the father diplomatically says, "You must have met the Spirit of the Forest." In an American movie, the father would have either accused the child of lying or been horribly patronizing.
These are the three main reasons why I found My Neighbor Totoro to be so charming, and such a refreshing breath of air.