How different is the working of SNN (Spiking Neural Network) as compared to a real Neuron System in biological systems?
There are different types of spiking neural network models:


Hodgkin-Huxley: models the processes within a neuron with electrical parts. This results in differential equations with 4 variables (capacity of the membrane, resistance of the ion channels, balance potentials, openings of the ion channels).
Leaky integrate and Fire (LIF): Probably the simplest SNN; it is only an ordinary first order differential equation 
Spike Response Model: models refraction time. It tries only to model the phenomena. Although it is simple, it is still more accurate than LIF.


I don't have a biology background, but I would say the Hodgkin-Huxley model is probably the closest to real neurons. However, as far as I know (which is very little), there is no (effective, plausible) training algorithm. So this is probably the key difference to the brain. Also the topology will certainly be different.

And, of course, we can model much less spiking neurons than we have natural neurons in a human. So the number of neurons is a key difference, too. I've heard SNNs have a few dozend neurons, probably up to several hundred neurons. The biggest MLP (CNN) models I've seen so far have about 150 000 neurons (see Deep Residual Learning for Image Recognition). The human brain has about 86,000,000,000 neurons (see Wikipedia).