Since both have characters and stories from traditional fairy tales, surely they can be considered fairy tales themselves. There are however, some key differences between Fables and Into the Woods and traditional fairy tales as well as between the two of them.
In Into the Woods, the basic story lines and endings of the various fairy tales are the same as the traditional ones. Also, there is a narrator as in many traditional fairy tales. The movie also opens with the line "Once upon a time." In Fables, the tales have been altered and they are in a modern setting rather than the setting in which they were originally written.
A difference that both fables and Into the Woods have from the traditional tales is the role of men in the stories. In both the men are not typical examples of princes from fairy tales. In fables for example, Snow White and her husband have split up because he was cheating on her. In Into the Woods, the Princes married to Cinderella and Rapunzel end up getting bored with their respective marriages and start to lust after new princesses and show a fickle nature not generally associated with fairy tale princes.
The differences in these representations and the traditional fairy tales from which they were adapted are I believe a reflection of the times that each were produced. Clearly, Into the Woods and Fables both have modern twists like the infidelity of the princes.