Here’s one important difference: the binary logic fallacy you describe arises from believing that two possibilities exist when there are actually three. It’s a species of what is sometimes called the fallacy of false dichotomy. The Tilde fallacy arises when someone believes that there is only one theory being proposed, when there are actually two. In the other three examples I discuss in the long linked paper, the unstated position is actually self-contradictory, which is why those arguments appear to work only when the arguer’s position remains unstated. The Atheist tilde fallacy is not self contradictory, but it does misuse Occam’s razor by falsely claiming that it’s position is simpler that the other alternatives.