Sibling rivalry tends to be pretty common, and the reason it manifests is that, like everyone else, children have a need for love and when they're competing for it, they can feel a lot of resentment toward the person they're competing with. To bring into perspective how that can feel, imagine if your significant other suddenly got another wife (in a polygamous relationship) and you had to compete for his love and attention.
Now, you mention that the dad left and has no interest in his daughters, which I imagine might make it even tougher for the girls. Now they're competing for the limited time of a single parent and they've gone through the trauma of losing one parent's love.
So here's how I imagine the 12-year-old girl is feeling: she NEEDS love and care from her mother, she's very angry that her sister is taking her mother's attention away from her, and maybe she's also afraid that her mom could lose interest in her, just like her dad did. When her mother withheld privileges, her girl might have took that as a sign that she loves her less, which I imagine might have made the resentment stronger.
How to deal with this? There's an article in Czech that I would recommend, but as it's in Czech, I'll try to summarize its recommendations (feel free to try Google Translate on it, though). When I say "you" here, I mostly mean the mother, though it's still applicable to you as the grandmother:
1) Sit down with the older child and let her express herself openly about the younger child. Don't judge. If she hates her sister, let her say that without passing judgment. And let her see that you understand her by repeating what she says ("You'd rather it be just you and me, huh?" or "So you really dislike it how much attention I give to your sister?").
2) When the older daughter acts out, try to empathize and don't punish or yell. Take her by the side and let her express herself, what she feels in the moment, and be empathetic. Once she's expressed herself fully, only then express yourself, but in a non-judgmental way — only by describing your own feelings (sad, angry, ...) and needs (peace, love, harmony, ...). The child should know that, even though you're not happy after what happened, you still love her.
(This might seem counterintuitive, but think back to the polygamous example. If you're really mad at your husband's other wife, the last thing you need is your husband yelling at you.)
3) Plan some regular alone time with each of the children, to let both know that they're still loved and to let them have some special time with the one they love.
Your daughter can try to do this alone, but sitting down with a child openly without judging can be tough to do in practice, especially when you're mad at them. So it might be better to go visit a good family therapist (make sure the therapist has a good track record) and let them mediate the interaction. That said, if your daughter doesn't want to go, then it's still worth it to try the solutions above.