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. For example, thinking of the word "fox" may automatically spread activation to memories related to other clever animals, fox-hunting horseback riders, or attractive members of the opposite sex.
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.
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).
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.
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.
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.
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. For example, despite your intuitive feeling that you could close your eyes and know the locations of objects around you, a series of experiments in the field of change blindness has shown that our visual memories are actually quite sparse. In this case, the brain is "offloading" its memory requirements to the environment in which it exists: why bother remembering the location of objects when a quick glance will suffice?
Bonus Difference: The brain is much, much bigger than any [current] computer
Accurate biological models of the brain would have to include some 225,000,000,000,000,000 (225 million billion) interactions between cell types, neurotransmitters, neuromodulators, axonal branches and dendritic spines, and that doesn't include the influences of dendritic geometry, or the approximately 1 trillion glial cells which may or may not be important for neural information processing.