Why Poisson Process for neural firing?
You can read about the differences between a binomial and poisson distribution at this other question on Math.SE.

First, since spikes are discrete events, it makes most sense to use a probability distribution that is discrete. A Poisson process is a process where an event occurs randomly with no "memory" for how long it has been since the last event. A Poisson distribution is the number of events if you integrate draws above some threshold from an infinitesimally small uniform distribution. Therefore, the Poisson distribution is the distribution you choose if occurrences of an event are only dictated by an underlying rate parameter - this turns out to be a reasonable, if ultimately false, assumption for neurons. 

There is definitely no biological fact that neurons must follow a Poisson distribution - in fact, no neuron can be truly Poisson because, if nothing else, the refractory period makes it impossible. Rather, Poisson behavior is sort of the simplest behavior you would expect from neurons in a system where they are firing occasionally due to random noise - sort of a "null hypothesis." For many modeling approaches, this is close enough, and it allows the modeler to avoid making more nuanced assumptions. It turns out to be a decent assumption when you look at individual cells, but if you look at populations the activity deviates a lot from Poisson.

Note that the Poisson distribution is often only valid for spontaneous data; it is possible to use a nonstationary Poisson distribution to model responses to some external stimulus, but the results are likely to deviate more from experimentally derived data.