There are a few differences:
-I believe Carrie in SATC actually claims to have a brother at some point?
-SATC changes the back story by saying her father left their family when she was 5, nothing about a dying mother, etc (the episode where she starts working at Vogue).
-Technically, in SATC II the movie it explains how & when they all met, which differs from the books where she meets one of the main ladies fairly soon after high school.
I find these differences kind of annoying since, story-wise, there doesn't seem to be a point to having changed them in SATC in the first place. I am even more annoyed that I can spot these differences so easily. They needed to stick with the books for the Carrie Diaries since they use her mother's death as a way to introduce her to NYC, but since this is a prequel it all gets weird when you do what all of us can't help but do: piece the story together in one line (which doesn't work).
Trying to take The Carrie Diaries as a separate piece of fiction & enjoying it for what it is--entertaining.