Both "Into the Woods" and "Fables" use stock characters that their audience will already know. They will already know the tropes, the peculiarities, the details of their story -- which is key for picking up on the inside jokes. That aspect of the stories, while very enjoyable, is something that is not seen in the straightforward plots of fairy tales. There are no sequels, no funny jokes outside of the satire we've seen in the last couple weeks.
I would argue that "Into the Woods" is a fairy tale in that there is the fantastical and the unhumanistic foe. There is metamorphosis with the witch, characters defying death, some magic, lots of quests, and lots and lots of advice/morals (although then those pieces of advice are spun around). There are lots of unexpected plot twists, which is neither fairy tale-like or un-fairy tale-like. There is quite a bit of character development for some of the characters, which is unusual for fairy tales, but overall I would say the story could be called a contemporary fairy tale, or a twist on the fairy tale (but still in the fairy tale genre).
"Fables," though, wasn't a fairy tale, in my opinion. It was a great murder-mystery, but there is nothing super fantastical. It's kind of like "Enchanted," only without the helpful animals. Fairy tale characters, sure, but not much of a fairy tale. It's set in a city, the characters are more or less normalized, the murder isn't even a murder. There's no quest, no foe. Just lots of people who don't like each other and have some baggage/relationship/family issues. The story doesn't even end with the sexy wolf getting the maiden. She pretty openly rejects him. That is definitely not in the category of popular ways to end a fairy tale.