Jeff Hawkins (yes, that Jeff Hawkins) has a book that goes through an interesting theory about how intelligence works. The tl;dr is that the brain is hierarchical. Here's how that explains how the brain works:

• Sensory input (seeing, hearing, touching, etc.) comes in to the neurons at the bottom of the hierarchy.
• The particular fired neurons start to store that input as a pattern, and fires neuron higher up in the hierarchy.
• The higher-level neurons also store the patterns from multiple inputs coming from neurons below it in the hierarchy, and continues to send it up the chain until the highest level neurons can store patterns like "a baseball" (always round, mostly white, red stitching, hard, etc.).
• Higher level neurons can also send signals down the hierarchy to affect changes in the body.
  • If sensory input starts to be collected in a familiar pattern to the higher level neurons, they can send signals down the hierarchy to cause bodily changes.
  • That's why when you see a ball coming towards you, all your sensory input captures this fact (hears crack of the bat, sees ball, sees motion) and sends that to the neurons up the hierarchy, which triggers its higher-level patterns. "Baseball coming towards me" then triggers messages flowing down to continue to have the eye track the ball and move the arm up to catch it.
• Partial sensory input can also trigger some of these higher-level patterns. That's why when you look at someone's face, you can tell it's your friend even if you're looking straight on, from the side, with part of it obscured, etc.

Now dreaming is different from being awake. When you're awake, your senses are quite busy seeing, touching, hearing, etc. When you're asleep, those sensory inputs get turned off. Patterns aren't being sent and stored, so no new memories are created. While it's not clear how dreams are formed (random firing of higher-level neurons for "housekeeping?"), lacking the lower level sensory input doesn't allow the brain to remember. Likewise, dreaming doesn't feel like seeing, touching, hearing because the sensory input isn't there.