Ugh. Being a teenager is hard. You're old enough to know your own mind but not old enough to act on it as you please. It's a real shame your mom won't allow you to express yourself in such a harmless manner.
You're a good daughter to care enough to want to makes things go smoothly with your mom. I was not like that. Had I had dreads as a teen and had my mother telling me to comb them out...I'd have done the exact opposite. Tension be damned. I'd had put as many dread extensions in my hair as possible. In every colour of the rainbow. Then I'd start doing other things like painting my face or stenciling fake tattoos all over my body.
You are at the age where there needs to be an expansion of mutual respect between parent and child. You might not be able to change your mothers mind but I think you should follow S.E.'s advice explain to your mother how you feel. Either in person or in a letter. I'm partial to letters simply because it's easier to explain yourself without the other person derailing the conversation or putting words into your mouth. Especially for such an emotional topic as this seems to be.
Your mom needs to know how you feel. Just don't expect her to respect your feelings right away. It seems she still views you as a child to mold and protect rather than someone on their way to adulthood.
Good luck to you. And remember that you are still beautiful. Your hair doesn't change that. You could be completely hairless and you would still be beautiful. Never ever forget that. Never.