It does not directly give you brain damage or cancer, that's a load of rubbish. Also, if it were true, they wouldn't sell it, and I'd be dead by now.
If you're worried, you shouldn't drink ANY soft drinks with the sweetening agent "Aspartame", it's in pretty much ALL sodas and soft drinks...as well as countless other consumer products. Prolonged exposure or consumption of anything can lead to cancer. Everything gives you cancer.