Atheists, an whomever else makes the claim, what is the difference?

I see no useful distinction. I had my doubts about the big bang until it was pointed out to me that this was an explanation for the presence of lithium and deuterium in the universe -- and, indeed, the only possible explanation. So, the proper approach to a proposition is that, if it is supported by evidence, and otherwise pes appropriate sanity checks, it is worth accepting/believing unless and until something better comes along.