Tolkien's Middle-earth is a land brimming with magic, so much so that the inhabitants (save a few) don't really care about it at all. We have elves who make a magic rope that will never break, and to everyone else it's special Elven rope; to the Elves, it's simply just rope. They have fantastic places like Rivendell and Lòrien that are swamped with magic, and thus are completely beautiful. They are wonderous places for, say, the likes of Bilbo, who would retire in Rivendell and listen to beautiful Elvish music until the end of his days; but to the Elves, they are simply called home.
Then we have the rings, those given to different races and, of course, the One Ring. Hobbits never received any rings, so when Smeagol finds the Ring he has no idea of the consequences, and neither does Bilbo when he takes it in "The Hobbit." All he knows is that it turns him invisible to everyone else, and is unaware of the danger that it brings. Fortunately for Frodo, Gandalf was there to warn him at the beginning of his quest. So while magic exists in Middle-earth, and pretty much all of its inhabitants know it, there are still certain magicks (like the enigma that is Tom Bombadil) that the people of Middle-earth have no idea about, despite the existence of other magicks that are so common that people don't give them a second thought.
Pullman's "His Dark Materials" trilogy matches up with these ideas quite nicely. While I wouldn't go so far as to say that the inhabitants of Pullman's world don't care about magic or its origins, at least most of them don't give a second thought to their daemon; it is just a natural thing in their lives that they have known since birth, just as Rivendell or Lòrien is to the Elves. The difference is that people in Pullman's story start to question their daemon's existence, as well as the origin of Dust. This curiosity leads, of course, to the discovery of other worlds, another comparable element to Tolkien. No, there aren't other "worlds" in his story, but the contrast between places like the Shire, Rivendell, Moria, Gondor and Mordor (not to mention where Frodo and the other ring bearers sail to at the end of the trilogy) is simply worthy of "otherworldly" comparison.
And, of course it goes without saying that both stories incorporate magical items (the Rings, the Golden Compass, the Amber Spyglass, etc.) but perhaps what is truly unique to the Pullman universe is the use of post modern technology. After much speculation, one could come to the conclusion that the Ring must be returned to the Cracks of Doom NOT because it was forged there, but because it is the only available source of magma in Middle-earth, and an abundant source of liquid heat is all that is required (just like the end of Terminator 2). This may not be true, but the fact remains that magic exists in the Ring, and to destroy it, simple technology will not suffice. In HDM, simple technology also will not suffice, particularly when separating a child from his or her daemon; time, money, and effort were put into building a machine capable of this feat.
Also, there are talking polar bears, but no other animals talk unless they are daemons. What's up with that?