Perhaps "not even close" was a bit strong, but I'm talking about a neural net as a specific mathematical model here. To say that the brain "is" an instance of a particular mathematical model isn't even really meaningful. At best you could try to argue that the brain is "modelled well by <formalism> for <purpose>", although I think you'd lose that argument for all but the weakest of purposes.

I'm not a neurobiologist (if you are please correct/clarify!), but here are just a few of doubtless many significant differences as I understand them:

* Real neurons don't update in a bunch of coordinated discrete timesteps, as an ANN learning algorithm does. They can fire independently and in continuous time.

* Real neurons' activation behaviour is closer to sudden delta-function-like spikes, rather than a smooth activation function or something which allows them to stay in a firing state for longer than a short pulse.

* The structure of the connection graph of neurons in the brain is incredibly more complex than that of artificial neural net models, can change over time, isn't split into obvious layers from input to output, there isn't a clearly-identifiable error signal which is used to train it at the outputs, ...

* A real neuron is an incredibly complex biological system whose behaviour can be modulated in all sorts of ways and which I expect would take a nightmarishly complicated set of PDEs to model with any degree of realism even as an isolated unit. To think we've captured all aspects of their behaviour relevant to human cognition with a simple weighted sum and a sigmoid (say) seems pretty naive.

Details aside though, the idea that you can start drawing deep philosophical conclusions about the nature of human thought, our ability to conceive of our own limitations etc, based on an analogy between a complex biological system and a simple mathematical formalism which is at best loosely influenced by certain limited aspects of it -- it's just silly, one of those "not even wrong" sort of statements.