What I think is the problem is that humans do not have a clear explanation for how our intelligence works. Is it really that much of a wonder we can't emulate it?
I agree we will likely not be calling these things "computers", if we ever invent them.
But a neuron is more than just an input-output state machine, it's affected by levels of oxygen, glucose, and any number of hormones, proteins, and other chemicals in the bloodstream. An adult human's neurons are each individually shaped by their entire existence up to that point. Alcohol consumption, sun exposure, antidepressant medication, hydration levels, exercise levels. It all affects how they work.
And that's just one neuron. Simulating the brain as a solution to this problem is, I think, out of the question.
Possibly, but why on Earth would you want to simulate that? Just simulate what the neuron is supposed to be doing or would be doing under ideal circumstances.
We still don't know that either.
You arbitrarily impose limits in the definition of a computer and lift those limits in definition of a brain.
On the other hand, "All problems in computer science can be solved by another layer of indirection." (http://www.dmst.aueb.gr/dds/pubs/inbook/beautiful_code/html/...) So if that applies to this problem (after enough levels of indirection) then maybe I'm wrong.
He develops this argument more than I have time to do here: http://www.wolframscience.com/nksonline/page-822-text (note that this is near the end of the main section of the book and rests on ideas established earlier).
Wolfram himself seems pretty bullish on the idea of making computers that think sometime in the future in the section I linked to, in a talk he gave at HAL's birthday party called Hal Isn't Here back in '97, and much more recently in some comments on the movie Her.
As for your question "what are those limits?": In a nutshell, no computer program can ever fully answer questions about the properties of other computer programs. Unlike what GP is implying, that is a hard limit we can never ever get rid of. So if weather is universal, we will never be able to fully understand it!
What Wolfram repeatedly points out throughout his book is that the threshold for such universality is much lower than one might suspect given the complications involved in, say, a Turing machine. And because that threshold is so low, there exists the possibility that much of the natural world that is not obviously simple is exhibiting universal computation.
1. Machine learning is moving more and more towards indirect programming i.e. you program the computer with a learning algorithm and let it work out what to do. Google reinforcement learning, or machine learning. This greatly reduces the programming bottleneck.
2. People underestimate how much processing power the human brain has. Think 100,000,000,000 neurons, each with 1,000 active connections on average and perhaps 10,000 latent connections (which are being updated via Hebbian learning). The connections (axons and dendrites) are the active processing units. The cycle time is .01 seconds or so. Only the very largest computers are anywhere near this processing power (~10^16 operations/second). My current desktop is about 10,000 times less powerful. Now imagine trying to build a tractor with a 1/100 horsepower motor - such a difference is beyond being a gap, it is a qualitative difference.
Given the limited processing power available it is amazing computers can do what they can. Back in the 1980s a large bank was run on the equivalent of (1/10 of a millimeter of brain tissue)^3.
Second computers are intentionally designed to be general purpose and they sacrifice a lot of potential speed to do this. If you were to build some algorithms into the hardware they would get vastly more performance (eg bitcoin mining), but that's extremely expensive. However being general purpose has a lot of advantages. Computers will always be faster at many things than neurons.