Although the brain-computer metaphor has served cognitive psychology well, research in cognitive neuroscience has revealed many important differences between brains and computers.
Difference # 1: Brains are analogue; computers are digital.
It's easy to think that neurons are essentially binary, given that they fire an action potential if they reach a certain threshold, and otherwise do not fire. This superficial similarity to digital "1's and 0's" belies a wide variety of continuous and non-linear processes that directly influence neuronal processing.
Difference # 2: The brain uses content-addressable memory.
In computers, information in memory is accessed by polling its precise memory address. This is known as byte-addressable memory. In contrast, the brain uses content-addressable memory, such that information can be accessed in memory through "spreading activation" from closely related concepts.
Difference # 3: The brain is a massively parallel machine; computers are modular and serial
An unfortunate legacy of the brain-computer metaphor is the tendency for cognitive psychologists to seek out modularity in the brain. For example, the idea that computers require memory has lead some to seek for the "memory area," when in fact these distinctions are far more messy. One consequence of this over-simplification is that we are only now learning that "memory" regions (such as the hippocampus) are also important for imagination, the representation of novel goals, spatial navigation, and other diverse functions.
Difference # 4: Processing speed is not fixed in the brain; there is no system clock.
The speed of neural information processing is subject to a variety of constraints, including the time for electrochemical signals to traverse axons and dendrites, axonal myelination, the diffusion time of neurotransmitters across the synaptic cleft, differences in synaptic efficacy, the coherence of neural firing, the current availability of neurotransmitters, and the prior history of neuronal firing. Although there are individual differences in something psychometricians call "processing speed," this does not reflect a monolithic or unitary construct, and certainly nothing as concrete as the speed of a microprocessor. Instead, psychometric "processing speed" probably indexes a heterogenous combination of all the speed constraints mentioned above.
Similarly, there does not appear to be any central clock in the brain, and there is debate as to how clock-like the brain's time-keeping devices actually are. To use just one example, the cerebellum is often thought to calculate information involving precise timing, as required for delicate motor movements; however, recent evidence suggests that time-keeping in the brain bears more similarity to ripples on a pond than to a standard digital clock.
Difference # 5 - Short-term memory is not like RAM
Although the apparent similarities between RAM and short-term or "working" memory emboldened many early cognitive psychologists, a closer examination reveals strikingly important differences. Although RAM and short-term memory both seem to require power (sustained neuronal firing in the case of short-term memory, and electricity in the case of RAM), short-term memory seems to hold only "pointers" to long term memory whereas RAM holds data that is isomorphic to that being held on the hard disk. (See here for more about "attentional pointers" in short term memory).
Unlike RAM, the capacity limit of short-term memory is not fixed; the capacity of short-term memory seems to fluctuate with differences in "processing speed" (see Difference #4) as well as with expertise and familiarity.
Difference # 6: No hardware/software distinction can be made with respect to the brain or mind
For years it was tempting to imagine that the brain was the hardware on which a "mind program" or "mind software" is executing. This gave rise to a variety of abstract program-like models of cognition, in which the details of how the brain actually executed those programs was considered irrelevant, in the same way that a Java program can accomplish the same function as a C++ program.
Unfortunately, this appealing hardware/software distinction obscures an important fact: the mind emerges directly from the brain, and changes in the mind are always accompanied by changes in the brain. Any abstract information processing account of cognition will always need to specify how neuronal architecture can implement those processes - otherwise, cognitive modeling is grossly underconstrained. Some blame this misunderstanding for the infamous failure of "symbolic AI."
Difference # 7: Synapses are far more complex than electrical logic gates
Another pernicious feature of the brain-computer metaphor is that it seems to suggest that brains might also operate on the basis of electrical signals (action potentials) traveling along individual logical gates. Unfortunately, this is only half true. The signals which are propagated along axons are actually electrochemical in nature, meaning that they travel much more slowly than electrical signals in a computer, and that they can be modulated in myriad ways
Difference #8: Unlike computers, processing and memory are performed by the same components in the brain
Computers process information from memory using CPUs, and then write the results of that processing back to memory. No such distinction exists in the brain. As neurons process information they are also modifying their synapses - which are themselves the substrate of memory. As a result, retrieval from memory always slightly alters those memories (usually making them stronger, but sometimes making them less accurate - see here for more on this).
Difference # 9: The brain is a self-organizing system
This point follows naturally from the previous point - experience profoundly and directly shapes the nature of neural information processing in a way that simply does not happen in traditional microprocessors. For example, the brain is a self-repairing circuit - something known as "trauma-induced plasticity" kicks in after injury. This can lead to a variety of interesting changes, including some that seem to unlock unused potential in the brain (known as acquired savantism), and others that can result in profound cognitive dysfunction (as is unfortunately far more typical in traumatic brain injury and developmental disorders).
One consequence of failing to recognize this difference has been in the field of neuropsychology, where the cognitive performance of brain-damaged patients is examined to determine the computational function of the damaged region. Unfortunately, because of the poorly-understood nature of trauma-induced plasticity, the logic cannot be so straightforward. Similar problems underlie work on developmental disorders and the emerging field of "cognitive genetics", in which the consequences of neural self-organization are frequently neglected .
Difference # 10: Brains have bodies
This is not as trivial as it might seem: it turns out that the brain takes surprising advantage of the fact that it has a body at its disposal.