Spirituality can be connected to the supernatural, and so can the gods. Thus, by rejecting the latter, atheists might have to reject the former.
I say this is a superficial implication. I'm aware of the idea of atheistic spirituality. In fact, the article goes on to show why atheists should be expected to be religious in a sociological sense. That's why behavioural "godlessness" might be broader than "atheistic belief."
But "spirituality" might be applied to atheism only because "spirituality" is ill-defined. In particular, atheism deprives spirituality of much meaning, once you remove gods from the equation. That's why I argue elsewhere that "spiritual" should be replaced by "existential."