I think  Carter is  rewriting fairy-tales because maybe when she was growing up she didn't like the way they all ending maybe she thought there's another way to these stories. her version are different from the original because in the original they all end "happily ever after", but  in Carter version they never seem to end to happy, someones always dies or the story end complete different from what you would expect.
for example in "The Courtship of Mr lyon" this story is similar to "The Beauty and The Beast" and in "The Beauty and The Beast" , when the beast was kissed by beauty he turned into a handsome man and they got married and lived happily ever after. but in "The Courtship of Mr lyon"  the tiger turned into a man with a broken nose, unkempt mane of hair and a heroic resemblance to the handsomest of all beast.