The nervous and endocrine systems control and regulate every other system. One way the nervous system acts as a control agent is by passing chemical messages between neurons via chemicals called neurotransmitters. These chemical neurotransmitters are passed from neuron to neuron at junctions called synapses. For example, acetylcholine is the neurotransmitter for skeletal muscle. When you are asleep you are not using skeletal muscles but you are using the muscles that assist in breathing (intercostals, diaphragm). So, when asleep, acetylcholine is not being passed from neuron to neuron via the synapses but other neurotransmitters are.
"Firing" is somewhat misleading in regards to synaptic activity. Because the activities of sleep and wakefulness are very different physically and biochemically , the exact same nuerobiochemical processes are not taking place.
First, synapses do not fire, neurons do. The synapse is the gap between the terminal buttons of one neuron and the dendrites of another neuron. Neurotransmitters are released from the terminal buttons of the initiating neuron, and they bind to the receptor sites of the receiving neuron. If the number of ion channels opened by the neurotransmitters is enough to depolarize the neuron then there is an action potential (i.e the neuron fires).
That said, neurons are specialized things, and it isn't really a question of whether there are different neurons that fire when you are asleep than when you are awake and more a question of what your brain is doing during your sleeping and waking states.
In fact, many of the neurons that fired during the day will again fire at night while your brain strengthens the connections you made (called long term potentiation) which helps with long term memory.
The more a neuron fires, the stronger the connection, and that is where the saying "to sleep on it" comes from. When you sleep or take a break after learning something you are far more likely to remember it than if you don't. If you study several subjects at once, and then rest, you are likely to forget much of what you did in between; however, if you take a break between subjects you are much more likely to remember.