To me, the most remarkable thing about recent neural network research is that a very substantial proportion of the human brain is dedicated to vision, and neural networks can now achieve human level performance at object recognition, and with similar, robust representations (see e.g.
http://journals.plos.org/ploscompbiol/article?id=10.1371/jou...).
I see two possible conclusions that can be drawn here. The first possible possibility is that neural networks really are close enough to the way the brain works, that they learn the way the brain does, and that with some algorithmic tweaks and the right training data they can learn to everything the brain does. This is the view that Luke is rallying against, and I tend to agree that it's too soon to make this claim. On one hand, the frontal lobe is not all that different from the temporal lobe. If we can achieve the same things with a machine as with one giant chunk of 6-layer neocortex, it's not too hard to imagine that we'll be able to do the same with other pieces. The problems may be different, but the biological implementation is strikingly similar. On the other hand, as sdenton4 notes elsewhere in this thread, the big difference between sensation and cognition is that recurrent processing is that the former can (apparently!) be accomplished in an entirely feedforward manner while the latter requires (or is at least thought to require for efficient implementation) recurrent processing. And just because we have efficient ways to train one kind of neural network doesn't necessarily mean they'll generalize easily to others.
But there is a second possibility, which is that neural networks don't learn or function quite like brains, but somehow both can find similar, nearly optimal strategies for encoding information in sensory domains. In this case, we're probably more than a few years away from machines that can think, but to us neuroscientists this is even more exciting. Vision scientists have been looking for decades to understand how the visual system works, and in this scenario, there's some underlying theory to vision and we have an easily observable system that implements it. Even if it turns out that vision is too different from cognition for these insights to generalize, the success of deep learning for object recognition still has vast implications for future understanding of the brain, and thus probably for future artificial intelligence. If we know the representations the brain uses, that makes figuring out the mechanism by which it computes those representations far easier than if we had to figure out both the mechanism and the representation from scratch.
So my perspective here is that we don't know exactly how far recent neural network research is going to take us, but it's hard to deny its importance. Before I started graduate school, I didn't think that machines would achieve human-level performance at object recognition before we figured out how the brain does it, but here we are. But I suppose that historically, huge advances in AI research have been followed by stagnation and disappointment, so who knows how far we'll get this time.