Our brains are only like neural networks in the most superficial ways. They're only called neural networks because of the way neurons look, not the way they function.Neurons alter their chemical composition for every process, we don't have one neuron for each memory, thought or instinct. Neural networks are not capable of altering themselves beyond tacking on more branches and even that is severely limited by what is already in the network.
This touches on one of the key reasons why neural networks are not AI. If a net receives input it can't identify it just throws up an error, ideally using it to improve its detection capabilities for next time. An AI needs to have the ability to identify something it hasn't seen before, even if only to identify it as an unknown entity, to be corroborated with any future examples. A neural net is only designed to identify the thing it's made to work on, so it just throws info on new items away in favour of improving its systems.