Why does Diet Coca-Cola float but regular Coca-Cola does not?

Because the high sugar content in regular coke makes it more dense. Diet sodas in general are less dense because the artificial sweetener used is much much sweeter than sugar, and so it is not necessary to add nearly as much (about 10 times less). So diet sodas are less dense than regular soda, and a can of diet soda will float in water but not regular.