In my teens, when I'd recently figured out that I'd been an agnostic for years, I thought that an atheist "knew" that there was no God, whereas an agnostic just couldn't tell and didn't see how everybody else was acting so cocksure.
Which is a reflection of the individual's personality, and doesn't have any bearing on the basic facts. I'm cocksure that 43*13=559, whereas someone else might not be so confident offhand. That I am confident and the other party is unsure does not make my position less correct, nor does the reverse.
I suppose there are atheists running around claiming they "know" that there's no god; but like you, I can't see how they have knowledge of a negative. (Unless, as someone else said earlier, they point out contradictions in the conventional definitions of god -- not terribly difficult, but not persuasive either). So there may be loads of personal definitions of "atheist," just as there may be several different kinds of "agnostics" around: (1) I don't know if there's a god; (2) I don't know if I'm really an atheist; (3) I don't know how I would figure it out; (4) I don't know much of anything; (5) etc. But I'll stick with the definitions I gave earlier. I think they're philosophically rigorous -- the difference between the atheist and the agnostic is their emphasis on the burden of proof.
No one asked, but personally I'm in my own category -- I don't know if the existence of the universe and the laws of nature can be considered evidence of anything outside of nature. So I'm hung up on whether there is any evidence, and I can't say definitely that there is or there isn't. I try to learn what science has to teach us. And I have no problem with religion, as long as it's not coercive. Benign religion is a social positive. Or so it seems to me.
I've seen the two camps divided into "strong atheist" (there is no god) and "weak atheist" (I personally have no supernatural beliefs). It's stupid, and I hate being classified as being in any "weak" category.