Not quite true, see lucid dreaming.

To be fair to the GP comment, I've personally never experienced lucid dreaming, so for me the distinction holds - when I'm dreaming, I never know I'm dreaming, and mostly I don't even remember my life awake, I only remember whatever's in the context of the dream.

That would be a third, distinct state from each. Dreaming while knowing that you are is quite a thrill!