D., I almost stopped reading after I read that you closed all your family's acts. Geeeez! This is between your mother and your sons. Why are you even involved? If the son that she didn't ask to be a friend wants to be a friend on Facebook then he can ask her to be a friend. If she doesn't respond, so be it. We cannot make everyone want us on Facebook. Facebook isn't even real life for most of us.
Also three weeks since your mother said she'd send a message to the other son is not very long. Did your son send an invitation to her? I think that your mother sent the invite to the one son because he has a new account. She may have not thought to do it when the older son got his. There are so many reasons for what happened other than her snubbing the oldest son. Most of us are not as detail oriented as you seem to be. And fortunately most of us are not so quick to become angry. It sounds like you were looking for a way to "get back" at her.
I wouldn't go see you either! Your post makes it sound like you and your mother are saying "if you don't do what I want, I won't do what you want." And you've reached a stalemate in which everyone is angry because the other one doesn't do what you think they should be doing.
You and your mother are having a snit fit like 2 siblings. Where is your love and respect for your mother? Where is your respect for your sons? Cutting them off is disrespectful to everyone involved including yourself.
Even if your mom is not an adult, you can be an adult. You might be surprised that eventually she will act more like an adult in response to your adult actions. You cannot change anyone but yourself. I do hope that you will realize how childish your post is and accept that you can do something about the way you feel and act so that you won't be so hurt and angry with your mother.
I also hope that you can allow your boys to have a relationship with their grandmother, based on their terms and without you interfering. They are teens and they will soon have to navigate in this world on their own. Now is the time for them to be learning how to handle this sort of situation in a calm and reasonable way.
I'm guessing that you've been hurt badly by your mother and believe that she doesn't love you. And she's been hurt badly by you and believes that you don't love her. The only way out of this hell is for one of you to start small and not respond in anger the next time. I suggest that you look for some counseling so that you can learn how to resolve your issues with your mother and become an adult who doesn't depend on her mother for validation.