Please consider giving this woman the benefit of the doubt. If you value her friendship -- her company, interests she shares with you, her sense of humor, her personality -- you might be able to see shades of gray in her actions and could be more forgiving and empathetic.
Some people are just group animals. If she "always" invites you and there are always other friends there, haven't you figured out yet that she is one of those people who prefers a group? You seem to view it as an insult or as her not liking to be alone with you and your kids. You know by now that others are likely to be there; do you ask her, "Is this outing just you and me, or will other folks be there?" Just learn from your experience and ask. I know people like her -- things aren't fun to them unless they're in a group. If you don't like that but want to see her, alone, ask each time and only say yes to an occasional group outing. Then take initiative to ask her places and say clearly, "This is a chance for just you and me to get together. I'd love to see you alone to talk."
Do you invite her to do things with you and the kids, or do you expect her to invite the kids--? Not clear to me. Does she herself have kids? If not, she may frankly feel awkward or overwhelmed by having kids around, or even may guiltily feel like she's in your way because you need to be keeping an eye on the kids when she's there taking up your time.
"When my sister died she didn't call or visit me or anything."
Yes, that's not great, and it's understandable it pained you. But a lot of people stick their heads in the sand when there is a death, even a death of someone a friend was very close to. The reason is not that they don't care or they dislike you; the reason is those people "don't know what to say" and fear that even expressing "I'm sorry" is going to come out wrong. It's not an excuse but it is an explanation. Many of us can get out an "I'm sorry about X's death" but there really are people who freeze up at that.
When her friend died, if you contacted her but she didn't want to get together -- she might have dreaded the idea of talking about something so painful to anyone, not just to you. Don't take it personally. Can you understand that some people grieve that way and don't reach out and don't want to talk, even if the person asking to talk has the very best intentions? Not everyone grieves the same way. I suspect from the two mentions of deaths that you are puzzled by how she reacts to deaths and you have different expectations of what people should do when there's a death. She may have a very different background and experience with this.
She hardly calls any more: Have you asked her why? When she doesn't call back, do you leave her a message or send a card saying you haven't heard from her and you are worried about her, concerned for her? If a friend didn't call I'd be worried before I'd be mad. And honestly it does sound as if she has lots of other friends; maybe she's simply very busy.
The birthday party complaint seemed to come from left field. Have you hosted a kids' party? At her own kids' parties she is the hostess and probably running around checking that all is well, games are going OK, kids are getting along, everyone has a drink, etc. Why be offended that the busy hostess is not making face time especially for you one on one? I was at a kids' party yesterday. Should I be offended that the hostess was busy and didn't talk to me beyond a few words here and there?
"Never invites me to her house." She may be like me. My house is loved but small, and consequently overstuffed. I also have work and kid duties that mean there sometimes is literally one area of the couch that's open for anyone to sit on. I try to keep up and we do have folks over but not as often as we'd like. So I meet friends at coffee shops or for walks (outdoors, or inside the malls when the weather's bad). We mom friends don't tend to go to each other's houses but to escape together to some nice "third place" like that. Why do you need to go to her house? Does she have others over but not you, or does she just not have folks over? This might be a reflection of the idea that she prefers going places and meeting a group while you would prefer seeing her alone at her home or yours. You just have different tastes and preferences but it's no reason to be offended with her.
"When I invite her places she cancels frequently. I sometimes have to cancel on some things, so I am not too offended by it."
So why was this added to the list of her offenses?