Hrm... I'd say atoms are real iff reality (whatever that is) obeys a certain set of (approximate) regularities (that basically amount to the rules for atoms). ie, there's a sense in which they're "actually there"

The atoms are an explanation for a phenomena iff the fact that those particular regularities are largely sufficient to explain the phenomena in question. That is, that one doesn't have to "dig deeper" to still explain things.

(Of course, if atomic theory in general had failed and only explained a single particular thing, that would suggest that the first criteria was violated.)