Is aspartame bad for your teeth, since it is not real sugar?

No it isn't but. It is not bad for your teeth but most of the time it is in drinks that are still not good. Diet soda still is very acidic and can still cause damage to your teeth.
Not aspartame, but.. The problem is not so much the aspartame as what food it is an ingredient in. Many drinks, regular or diet, are actually very acidic and will erode enamel. Many foods with the sweetening agent are snack foods consisting of simple carbohydrates that are converted into plaque in the mouth. All in all, not so good for your teeth!