If you want to be treated like an adult, then act like one.
First, figure out how you can make the money and pay it back to your mother. You could baby sit, do yard work, make bake goods and take them to a soccer field when there are a bunch of games on, etc. There are TONS of ways that you could get focused and pay your own debt.
Then you could keep earning your own money and use it to pay for things you want instead of asking your mom. Besides proving how mature you've become, it would give you more control over your own life. It's really nice when you want something to be able to just buy it instead of asking someone to give it to you. One of the best ways to prove maturity is self-efficiency, even if it is just for movie tickets and cute clothes.
Second, sincerely apologize to your mother. Accept the fact that in that moment, your behavior was completely unacceptable and humiliated her. Don't even mention your brother. This isn't about him. Take responsibility for what you did. Understand how she felt. You shamed her, and people judged her based on your actions. I know you aren't quite here yet, but getting here is really important for your relationship with your mother. Part of being mature is being able to see something from another person's point of view.
Third, drop all whining about the x-box, drivers ed, ect. I suspect that if you focus on getting your own act together, the things you want will flow back to you over time. But you screwed up, and now you are paying the consequence. So you need to fix it, and then prove that you have matured through this experience.
This is related to learning to drive. Learning to drive is a tremendous responsibility, and clearly, since you cannot handle having access to ketchup without ruining something that belongs to someone else, you aren't ready to be behind the wheel of a car, which has a far greater ability to destroy and hurt.
With all your extra time (since no x-box, etc), completely ace your school work. Think about what you want to do after highschool, and make a plan for your highschool classes based on your plan. If you have no ideas, meet with your counselor at school and talk to him/her. Planning for the future is another thing that shows maturity.
Being a grown up isn't about getting to do what we want to do, it is taking full responsibility for our own lives.
You have a tremendous amount of control over your life right now.
Use that control to create a good life for yourself.
Also, ignore your brother. Walk away. Hide in the girls bathroom. Anything but engage. Don't sink to his level, because he will bring you down.