Do you get the same amount of rest from sleeping as laying down with your eyes closed?

No. Sleep is a beautifully orchestrated, highly conserved, complex body function with several brain systems working in concert. It is a vital restorative process that among other things helps with memory consolidation, metabolic function, immune competence, and growth. Quiet rest does not achieve the same brain state as sleep. Sleep is just as important as diet and leading a non-sedentary lifestyle.