They're different in practically every way. It's easier to enumerate the things they have in common: NNs borrowed the idea of using a connected network of functions whose outputs feed into each other's inputs.
That's it. That's the total resemblance between the two. The brain isn't just an NN implemented in biology, it has whole systems that aren't accounted for in digital NNs, like hormones and neurotransmitters, and even the system of connected neurons doesn't work the way digital NNs implement it.
Neural networks model the brain exactly as well as objects in OOP model cells: not very well at all. They're inspired by biology, nothing more.
Neural networks as used in AI are inspired by the brain in much the same way that OOP was inspired by the way cells work--neither one is an attempt to faithfully recreate the actual operations of an extremely complex (and only partially understood!) biological system.