The two processes are comparable, but our unconscious thinking is freer. Try the following experiment. Take up a tough math problem or word puzzle and try to solve it, and while working on the problem, observe when you switch from conscious activity to unconscious activity. I think you will find that you will work consciously for a time as long as you feel you have an approach that might work. When you lose faith in your approach, however, your mind will go blank until a new "idea" emerges, which will reveal itself as being a new organized approach to try. You will continue working on this until it shows itself as being still not quite right. At that point your mind will go blank again, and this switching between conscious and unconscious work will continue until you finally hit upon an approach that through conscious work will finally bring you the correct solution.
The reason our minds switch back and forth is that our unconscious processes are freer to stray beyond the boundaries of logic. We don't get insights into problems using logic alone. Our creative processes are a bit mysterious, but it's known for certain that something besides logic is involved. Logic is something we add at the conscious level after an unconsciously derived approach has been worked out.
Let's now take the case of a really tough problem. In this instance, we probably have no idea initially how to put together all of the information at hand to arrive at a solution. So we find that we cannot initially muster any conscious thoughts at all; we must sit there squirming while our minds remain resolutely blank. Then after a long while maybe an idea will pop up and we will start working consciously. But if it's a really tough problem, this approach will get shot down right away, and our thought will be driven to the unconscious level for another long time. Generally speaking, the next conscious approach will be better than the first, and thus permit a longer period of conscious work. As we go on, our unconscious activities will become shorter while our conscious activities grow longer until we finally solve the problem at the conscious level.
What am I getting at? I'm getting at how I came to see that sleep must relate to problem solving. Go back to my Black Boxes monologue and you will find that I characterized both REM dreaming and conscious thought as experienced thought in a literal sense.
When we first go to sleep, we have no insight into how to put together the information we've just gotten from the day's experiences, so sleep begins with a long purely unconscious interval that ends with a short experienced REM dream. That dream is the result of an idea for a solution that peters out pretty quickly. We then go back to purely unconscious thought, which results in another idea. This one's a bit better, generally, so the second REM dream lasts a bit longer. Throughout the night, our REM dreams get longer at the expense of our purely unconscious sleeping intervals as we get a better handle on how the new information we have received should be incorporated in our life.
We can no more solve tough problems purely at the conscious level than we could sleep gainfully using REM dreams alone. We need unconscious thought, and the reason is that it's freer nature allows for more creativity.