Do the same synapses fire when you are asleep and when you are awake?

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.