(NOTE: These are mostly my ideas, cribbed from my reading. I am definitely open to discussion on this!)
Fantasy grew out of fairy tales sometime during the Victorian period to become its own genre, although elements of fantasy have been around as long as there have been stories. Something like The Odyssey has very strong fantasy elements: monsters, witches, etc.
Here’s the defintion of Fantasy and its traits from Wikipedia:
is a genre of fiction that commonly uses magic and other supernatural phenomena as a primary element of plot, theme, or setting. Many works within the genre take place in imaginary worlds where magic is common. Fantasy is generally distinguished from the genre ofscience fiction by the expectation that it steers clear of scientific themes, though there is a great deal of overlap between the two, both of which are subgenres of speculative fiction…
The identifying traits of fantasy are the inclusion of fantastic elements in a self-coherent (internally consistent) setting, where inspiration from mythology and folklore remains a consistent theme. Within such a structure, any location of the fantastical element is possible: it may be hidden in, or leak into the apparently real world setting, it may draw the characters into a world with such elements, or it may occur entirely in a fantasy world setting, where such elements are part of the world. Essentially, fantasy follows rules of its own making, allowing magic and other fantastic devices to be used and still be internally cohesive.
So fantasy usually takes place in some sort of “other world”, that is not our own. Fairy tales exist in a world that we know, but a long time ago, when magic was still possible, fairies existed, etc.
Tolkein and C.S. Lewis’ works are fantasy, not fairy tale, and could even been considered allegorical. The Lord of the Rings and the Chronicles of Narnia follow their own internal rules, landscapes, and societies. Lewis does give us a bit of the “real world”–the London of World War II–but Narnia is completely separate from any sense of being “real world”–as in, existing with London. It’s a parallel universe. Neverland is like that as well, and so is Oz. The places that Dorothy, Wendy, Michael and John go to are real , but they don’t exist in what could be considered “our” world.
Also, these people tend to be people like us. Dorothy is just a little girl from Kansas. Wendy is a girl from London. The Pevensie children aren’t royalty or “special” in any sense of the word. Frodo is definitely quite a normal little Hobbit until Fellowship begins. As we’ve seen, in Fairy Tales, the characters are usually royal or peasants or something fantastic.