What is the difference between different brain regions
The wiring is different as you mentioned. However, perhaps the most important is the brain knows where it's input is coming from. The brain knows where each fibre innervates and thus can compile and present this data to our conscious mind. We show if we stimulate the brain directly, than we feel a sensation in the part of the body that portion of the brain is responsible for. Plasticity means our brain can change what feeds into where, this is most commonly where we learn a motor skill. If we play the piano for example the part of the cortex feeding to this area increases. 

Another way is the type of chemical transmitter and receptor. Dopamine is primarily used for things that cause us pleasure for example. However dopamine can affect our movement if it is secreted in from the substantia nigra, as this feeds into the motor cortex. Furthermore, neurotransmitters can be excitatory or inhibitory and this is an analogue rather than binary signal. All of these fine tune, and the position at which they inhibit and the location and the feedback from which they obtain their signal all indicates from where the brain is getting the information or to what it is responding to. 

In summary it is wiring, signalling and location. However the components are incredibly similar but it isn't the small differences that have a profound effect.