I would say the biggest reason to avoid aspertame/nutrasweet is that it is a chemical. It is thought to possibly cause cancer, and it definately causes digestive problems. If you have read the Atkins book, or the Atkins age defying diet, he mentions that NutraSweet was originally developed as a PESTICIDE, until someone accidentally discovered it was sweet. Gross.
I try to avoid it, I use Stevia and Splenda instead. I only wish that diet sodas and gum were made with splenda rather than nutrasweet.