What is the subtle distinction between probability = 0 and "impossible"?

I asked someone that once

What if something has probability zero and it happens anyway?

He said Maybe it means it'll never happen again.

The subtle distinction is that if something is really impossible then you might not know it. And if you have some situation where something really has probability zero then it is due to a mathematical model. The mathematical model might be correct or not correct for a given situation. It gets into issues of applied mathematics vs. pure mathematics. If you are doing pure mathematics then something with probability zero will never happen, end of story. In any real world scientific application all the models are approximate. It comes down to a profession's ability and willingness to understand and represent reality accurately. It usually has some compromising factors like the requirement that the profession should be maintained as a profession, and that it should be systematic so its practitioners don't have to go to heroic measures to figure out a problem exactly. Research and applied professions are separate worlds and so is pure math.