Developed by Erik Rabkin in his Fantastic Worlds: Myths, Tales, and Stories (Oxford UP, 1979 ISBN 0-19-502541-5). The book pulls together a number of excellent (and relatively short) examples of these three story types.
Myth: creates, establishes and explains the world. Myth is oldest form of story. Some philosophers (Ernst Cassirer, for instance) have even argued that myth and language originated at the same time in one and the same process. Myths are believed to be true by those who first tell them.
Folktale: As cultures become more sophisticated and more conscious of the artistry behind their myths, they begin to intentionally shape examples of those myths to offer up as satisfying entertainment, with only a secondary intent to establish and explain the world. These are corporate productions of the whole culture. Folktales are taken as significant but, unlike myths, are admitted to be made by people.
Fairytale: As culture get even more sophisticated, a special set of narrative conventions can grow up which will attach to one class of stories -- and its associated audience -- or another. Examples are science fiction, modern fantasy, horror and the fairy tale.
Myth, folktale and fairy tale form a sliding scale along which stories become:
1) more conventionalized
2) the audience becomes more limited
3) the teller becomes more sophisticated
4) the truth value becomes more symbolic and less literal
BUT, in all three story types, the issues remain the same