Furthermore, the brain is massively parallel, and has a specialized architecture. For what it does, it's well optimized, at least compared to how optimized our software and hardware are for similar tasks at this point. For instance, laptop processors are general purpose processors, being able to do many different tasks they aren't really fast or good at any.

Intuitively this doesn't seem right at all: I can think of plenty of things that a human plus an external memory aid (like a pencil + paper) can do that a laptop can't, but (aside from dumb hardware stuff like "connect to the internet" and so on) I can't think of anything for which the reverse is true; while I can think of plenty of things that they both can do, but a laptop can do much faster. Or am I misinterpreting you?

It took 35,000 processor cores running to render Avatar. If we assume that a Six-Core Opteron 2400 (2009, same year as Avatar) has roughly 10^9 transistors, then we have (35,000/6)*10^9 = 5.83*10^12 transistors.

The primary visual cortex has 280 million neurons, while a typical neuron has 1.000 to 10.000 synapses. That makes 2.8*10^8*10^4 synapses, if we assume 10.000 per neuron, or 2.8*10^12.

By this calculation it takes 5.83*10^12 transistors to render Avatar and 2.8*10^12 synapses to simulate something similar on the fly. Which is roughly the same amount.

Since the clock rate of a processor is about 10^9 Hz and that of a neuron is 200 Hz, does this mean that the algorithms that our brain uses are very roughly (10^9)/200 = 5000000 times more efficient?

The switching rate in a processor is faster than the firing rate of neurons.

as a rough estimate it is reasonable to estimate that a neuron can fire about once every 5 milliseconds, or about 200 times a second

All else being equal, a computer should be faster than an aggregate of neurons. But all isn't equal, even when comparing different processors. Comparing transistors in a modern processor to synapses in a human brain yields many more synapses than transistors. Furthermore, the brain is massively parallel, and has a specialized architecture. For what it does, it's well optimized, at least compared to how optimized our software and hardware are for similar tasks at this point.

For instance, laptop processors are general purpose processors, being able to do many different tasks they aren't really fast or good at any. Some specific tasks may make use of custom made processors, which, even if their clock rate is slower, or if they have less transistors, will still vastly outperform a general purpose processor if they are to compete for the task they were custom-built for.