Sadly, I know how you feel.
I had what I thought was a very close friend that I was planning on going out of state to visit. We had a disagreement about a week before my mom, kids and I were supposed to leave.
When I called said friend to discuss the issue, she hung up on me and began posting rude general statements about me and my family, which was then echoed by her teenaged children and left "unchecked". We decided at that point to not go. The day after our supposed departure date, we got a serious health diagnosis for our daughter (a rare form of cardiomyopathy) and all was forgotten on my part... but the public war was still waging on her end. We were definitely not going and it didn't matter why at that point. She "apologized" by instructing me on forgiveness according to my faith, not hers, several times and I said I was sorry for stuff that I didn't think I should have to be sorry for, but was because of the way she said it made her feel. Eventually, I just wrote to her and said that I didn't know how to get past the harassment and public display of disdain.
I then received a nasty letter from her calling me names, claiming that she had apologized (nowhere in her correspondence with me up to this point contained the words "I'm sorry", but rather quotes of scriptures instructing me on forgiveness) and outlining everything she hated about me and thought made me less of a person than she is... things that I had confided in her beforehand and she had kept in confidence and supported me in until now... things like struggles with the schools, how we handle our kids' discipline, minor marital discourse, the one time that I forgot to pay a bill and my mom helped me out and I paid her back, even the fact that I was refusing to take my daughter far from her doctors because of this new cardiac diagnosis that we knew very little about. I now know exactly what she thinks of me, my husband, my children and even my mother... none of it's pretty.
The sad part is that I keep trying to look past all of that and still have some semblance of a relationship with her because she is married to my brother. It's very hard to be friendly with someone who I know thinks so little of me. I feel like she lied about who she was and used everything that I had shared with her about myself as a weapon to hurt not only me, but also my children... to whom she made personal attacks on in that letter.
My advice to you is to work to rise above the garbage, show her common courtesies, and respect the fact that she doesn't want her daughter to be friends with yours because of your falling out. Explain to your daughter that it's not her fault and you don't think it's fair, either, that she has to be punished for something that should have been handled differently.
If this other woman has an issue sharing room mom responsibilities with you, she needs to be the one to complain to the teacher and request a schedule change or withdraw completely. Ending the friendship and having bitter feelings has been her decision.
Finally, let it go. Just let it and her go. You can be neighbors with someone and not be their friend.
Eventually, someday, she will start talking to you as if nothing ever happened... that's what my sister-in-law did. Maybe just giving her time will allow her to come to her senses. There's nothing to say that when she starts talking to you, you then have to become her best friend.
Not everyone is going to like you and to lose a friend over a minor difference of opinion tells me that she really didn't consider you that close of a friend in the first place. You're allowed to decide to keep her at arms length if she wants to come back into your life... or even further if that's what you want.