Humans have been trying to cheat nature from the very beginning. Aspartame could be seen as the crowning jewel of that quest, or the shat-upon failure of the scientist who originally designed it to be an anti-ulcer medication.
So many random health problems go unexplained these days that people will go out of their minds trying to find something to pin them on. Aspartame falls neatly into the category of "it has a warning label that I don't understand, so it's dangerous right?" and therefore is perfect fodder for these people. An armada of lobbyists constantly bombard the FDA with dubious claims of massive tumors in rats and other carcinogens present in Aspartame despite the fact it only has two ingredients, aspartic acid and phenylalanine, both of which are amino acids naturally present in your body.
Aspartame is one of several artificial sweeteners available. Sodium Saccharin and Sucralose are its main competition. It's surprising that people are going apeshit about Aspartame when you hear about how ridiculous Sodium Saccharin is. It was discovered by a chemist at Johns Hopkins who was working on coal tar derivatives. In the decades since, the USDA has made numerous unsuccessful attempts to ban its use in America. Sucralose is tame by comparison, despite being 600 times sweeter than sugar.
If it's safe, why does that woman look so shifty?
The success of sweeteners like Aspartame, Sucralose, and Saccharin really outlines what a bunch of lazy pricks we are. We can't be bothered not to eat real sugar. The media is heavily overpopulated with commercials for pills and programs promising weight loss saying that you won't have to exercise or change your diet at all, despite such claims making no logical sense.