I think what you did in not inviting her to your Christmas party was fine. I don't see it as a "you did it to me, now I'm doing it back" but rather realizing from the not-so-subtle hint of not being invited to the birthday party 2 years in a row that the neighbor girl does not think of your daughter as a close friend.
Sometimes just because kids are neighbors and they go to the same school doesn't mean they are good friends. It sounds like your daughter might be a friend, but not a close one. It sounds like your daughter is good enough to be a playmate when the neighbor girl is bored or when she's invited over, but not good enough to be #1 on the list when it comes to having a friend over.
My daughter is 10, and when we tell her she can have a friend over she has one or two friends who are at the top of her list. She has a few friends that are not so high, and if her "favorites" can't play then she'll ask the others. She will, however, go over there if invited. I do try to reciprocate playdates no matter who the friends are and of course I invite ALL of her friends to her birthday parties. But you know, some moms don't care so much and don't really pay attention.
You know, it sounds like perhaps the daughter didn't want the child at her birthday party, NOT the mom. And perhaps when the Christmas invitation didn't come the mom told the daughter "well, you didn't invite her to your birthday party so she doesn't have to invite you to her Christmas party" and that's where the apology came from.
I would let your daughter lead you on this one. If she wants to invite the neighbor girl over then let her. If she wants to invite her to parties, let her. If she doesn't, then she doesn't. And if the neighbor mom brings it up to you then you can just say "I was just under the impression that the girls weren't that close, even though they are friends."
Life's lessons are hard, but perhaps the neighbor girl learned one.