How is interference between photon wavefunctions within the Hong-Ou-Mandel effect different from interference between coherent laser beams? Does it have anything to do with the difference between Fock states and coherent states? Do photons also exit together on the same output port of a beamsplitter in the case of interference between laser beams like in the case of Hong-Ou-Mandel effect or is it a different situation?
 A: The interference that one can find in a coherent laser beam can be described purely classically. On the other hand, the type of interference described by the Hong-Ou-Mandel (HOM) effect is a quantum effect. The latter involves multiple particles, whereas the former can be decribed with only one particle, one photon.
With the HOM effect one has two photons entering the two input ports of a beam splitter. Let's call them $a$ and $b$. Let's assume the two photons are identical (same polarization, same frequency, same spatial modes). Then we have the input state given by $|1\rangle_a |1\rangle_b$. The beamsplitter is described by a unitary process. One way to represent this process is as follows (assuming we can also label the two output ports as $a$ and $b$):
$$ |1\rangle_a \rightarrow \frac{1}{\sqrt{2}}(|1\rangle_a - |1\rangle_b) $$
$$ |1\rangle_b \rightarrow \frac{1}{\sqrt{2}}(|1\rangle_a + |1\rangle_b) $$
So, now it is clear to see that for the given input states (two identical photons) one would have
$$ |1\rangle_a|1\rangle_b \rightarrow \frac{1}{2}(|1\rangle_a |1\rangle_a - |1\rangle_b |1\rangle_b) = \frac{1}{\sqrt{2}}(|2\rangle_a + |2\rangle_b . $$
This means that either both photons leave through port $a$ or both leave through port $b$. The case where one leaves through $a$ and the other through $b$ is canceled by this process of quantum interference. Those terms appear with opposite signs and therefore cancel. In a sense the interfere destructively. This is different from the interference found in laser beams.
It does not have anything to do with being described by coherent states, as opposed to Fock states, because for optical interference the result is the same regardless whether one uses Fock states or coherent states. For instance, considering a one-particle Fock state (single photon) going through an interferometer, one would have
$$ |\psi\rangle= \frac{1}{\sqrt{2}}[|1\rangle_a + \exp(i\phi) |1\rangle_b] , $$
where the subscripts again represent the two input port to beam splitter that would combine the two paths of the interferometer and $\phi$ is a relative phase due to a difference in pathlength. Now we apply the operation of the beam splitter
$$ |\psi\rangle= \frac{1}{2}[(|1\rangle_a - |1\rangle_b) + \exp(i\phi) (|1\rangle_a + |1\rangle_b)] . $$
To see the interference we measure the probability to detect a single photon after one of the output ports. For this we use a projection operator $P_a$, which the selects the part after port $a$ 
$$ |\psi_a\rangle = P_a|\psi\rangle = \frac{1}{2} |1\rangle_a [1+ \exp(i\phi) ] . $$
The probability to detect a single photon is then given by 
$$ |\langle 1_a|\psi_a\rangle|^2 = \frac{1}{4} |1+ \exp(i\phi) |^2  = \frac{1}{2} [1+ \cos(\phi)] . $$
Here we see the interference as a function of the relative phase. One can see that this is quite different form the situation with the HOM effect.
