My best approach is to research this William and my William--just so I can keep them straight. I may research "my" William in more detail and in a more exhaustive nature, but I need to have enough material on the "other" William so that I can determine (as best as possible) which one is which. It will turn out that both these Williams were from the same region of Germany and spent time in the same area of Illinois.
They do that on purpose, just to confuse us.
Well, not really. The problem is that in cases of immigrants, individuals with the same name may be unrelated but, because they share a common heritage, language, and culture, may live in the same areas and interact with the same people--making them a little more difficult to separate and a little easier to confuse. That's what they do on purpose and sometimes that does confuse us.