To be normal is to adhere to a norm, meaning a set of either ethical or aesthetic standards, the following of which ensures, if not a higher moral or aesthetic ground, at least the avoidance of a lower moral or aesthetic ground. The antonym of "normal" is "deviant," meaning to deviate from the desirable norm.
By contrast, to be "typical" is to be an example of a type, which confers no moral or aesthetic superiority: one type can be as good as any other type, so that to be "typical" is simply to be conforming to the type. It also implies that one can move among types, identifying with one type one day, another type the next, without experiencing any moral or aesthetic decline. The antonym of typical is atypical, which simply means not of the type, without any moral or aesthetic judgment attached.
Herein lies the difference between the concepts of being "normal" and being "typical": you get to define what is acceptable and what is unacceptable, what is the type you want to be and what is the type you don't want to be. It makes us to be like gods, knowing good and evil--to be more precise, to think that we are like gods, deciding for ourselves what is good and evil. And that is precisely where Satan likes us to be.
Obambam used to talk about “norms” all the time. None of us is “normal” compared even to pre Industrial Age people. So the only constant is change. If we can simply focus on treating each other with respect, we would be doing well.