"Neural Network", tragically, is a term that refers to a specific kind of model inspired by a sort of 70s understanding of human neurons. Deep learning refers to those models. The human brain is made of neurons, and by God do they network, but the term has been co-opted so it is not a neural network. A particular region or set of neurons that do one thing might be called a neural circuit, and the whole thing can be called the connectome (though that's sort of debatable).