A fable always has a moral or lesson. A fairy tale is full of magic and usually contains magical numbers (3, 7) and a plot of good versus evil. The previous statements are generalizations for these two types of tales however each of the rules can and have been broken due to the author's wish. Just because fable's always have a moral or lesson does not mean they are not filled with magic and fantasy. The majority of fables have animals that are main characters and are extremely personified. Is this not a mystical element that has been added to the "fable reality"?
Into the woods takes the characters from multiple fairy tales and elaborates a story in a fantastical way. The general consensus would state that it has all of the necessary elements to be categorized as a fairy tale, if the definition is a simple as being mystical and fantastical. By including the fantastical characters from previous stories written while also following the traditional good vs. evil theme, yes into the woods could be categorized as a "fairy tale". If we all truly know what that is.