"Known unknown" vs. "unknown known"
The full quote is:
[T]here are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don't know we don't know.
That explains what known unknowns means: we know there are some things we do not know. As for unknown knowns, a philosopher by the name of Slavoj Žižek extrapolated to define this term: the things that we know, but are unaware of knowing.
So, in short, there is a big difference.
The quote is:
[T]here are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don't know we don't know.
"Known unknown" implies there are things we know we don't know, while "unknown known" could imploy things we know but don't yet realize the value. Thus, there is a difference in meaning.
Some people were keen to mock Rumsfeld's words, but they really are simple and easy to grasp. To avoid misunderstandings about any of the three combinations he actually used, Rumsfeld defined each one very succinctly immediately after saying it. I won't bother repeating the known bits.
You don't often hear the combination he didn't say – unknown knowns – because we find that one hardest to conceptualise. But I suggest that we have a particularly good example close to home here at EL&U. We all "know" a lot more about the principles governing correct usage of our mother tongue than we're conciously aware of.
This from "In Sleep", in William Logan's Sad-faced Men (1982)...
The stars madden, and satellites hum silently
Where no sound can awaken a sleeper. She dreams
A language which cannot trouble her, a vocabulary
Without voice. She has forgotten the ugly grammar,
The deformed sentences. She has forgotten
The irregularities of memory, the unknown knowns,
The can-no-longer-remembers, the slow impeachment
Of experience. She rises, stunned and serene,
Toward the promise of nothing.