What makes a human processing information so different from a set of instructions in a computer? Solving a problem for any human operating with concepts is still much superior to lots of computer processing. I fail to verbalize what this 'extra' capability is. I just see the result when observing humans solving problems, but what is the mechanism that allows humans to process differently.
I'm taking your question to be equivalent to "how does the human brain differ from a computer?". Indeed, it's well-established that humans outperform computers in a large number of contexts, but it's difficult to pinpoint exactly why this is. The best answer you'll get is one that outlines how the two computational systems differ.
Let me start with three obvious differences, which IMHO are the most important:
Brains are analog machines, whereas computers are digital. This means that brains can apply any number of continuous, non-linear transforms to solve problems. If you're mathematically or programmatically inclined, here's a wonderful paper that demonstrates how a brain can elegantly and efficiently discriminate between two stimulus frequencies (supplementary materials here if you want to mess with the code yourself).
Brains are good at fuzzy-logic and uncertainty. Contrary to computers, brain activity can "evoke" or "activate to varying degrees" similar representations. Intuitively, you can think of this as local activation spreading to adjacent areas (although this is a gross oversimplification). On a more cognitive level, this enables things such as semantic priming to be implemented quite naturally, whereas this is horrifyingly complex with digital computers.
Synaptic connections are much, much, much, much, much more complex than logic gates. Synaptic connections can be reinforced and inhibited on either end of the cleft (presynaptic or post-synaptic). Moreover, the dynamic nature of synaptic connections allows for some interesting emergent properties such as mutual inhibition, or the formation of multiple iso/nullclines in firing rates via more complex interactions (see the Machens et al. paper I linked to in point 1).
With this having been said, let me reiterate the point I made in my previous comment. Human brains have cognition, in the true sense. This alone makes them able to solve problems that other systems can't. To use your own words, humans have conceptual reasoning, whereas computers do not... which is to say the answer was in your question to begin with.
To expand on what is admittedly a not-very-useful last paragraph, it seems as though you are conflating concept and representation. A concept implies conceptual knowledge, which is simply non-applicable to a non-cognizant system. As to how the representations differ, this is where point
2 is relevant. In this case, the representation of a face evokes activation corresponding to various related representations (e.g. on the semantic level). Importantly, this "spillover" occurs by the very nature of the system -- it's not something that's artificially tacked on.
It's the relationship between your representation, other representations, and the sensorimotor system ("self") that arguably constitute a concept.
Fletcher, P. C., Shallice, T., Frith, C. D., Frackowiak, R. S. J., & Dolan, R. J. (1996). Brain activity during memory retrieval The influence of imagery and semantic cueing. Brain, 119(5), 1587-1596.
Machens, C. K., Romo, R., & Brody, C. D. (2005). Flexible control of mutual inhibition: a neural model of two-interval discrimination. Science, 307(5712), 1121-1124.
Intuition is what distinguishes human intellect with classical computers. The human intellect is intuitive in the first place and it builds strict logical (conscious) mind on top of it. Computers are opposite. They act according to concrete algorithms. Yet, you can achieve intuition by emulating it with computer logic. There is nothing that prevents machines to become intellectual.
I you want details, the intuition is achieved through connectionism (Neural Network), a computational model alternative to the hardwired logic of computers. The Neural Network is ultimately parallel computer. It consists of utterly stupid computational units, neurons, every is connected with (tens of ) thouthands of other neurons. Connection pass simple pulses, raised by neurons. The reaction of the brain depends on the weights of these connections (synapses). Learning is basically adjusting of these weights. This is one of the strength of intellect over computer -- it programs it itself, by the environment, which allows to make decisioins for the problems which are intractable to formalize and it guesses the best answer in the abscense of complete information and does so immediately (in matter of couple of neuron pulses, in fractions of a second), thanks to the immense parallelism. These are the things unattainable by stupid, programmed sequential algorithm. I have told that parallelism is achieved through the myriad (100 billion neurons x tens of billions of connections per each = quadrilions of processing units in one head) of connections, working in parallel. You thus may guess that even Blue Gene has problem to compete with it with the tasks that are difficult to formalize or which have large search space. You may also notice that parallelism of NN is squred by the fact that every synaps holds all the images exposed to the brain before. Yes, every element holds all immense informatioin known to you and you have quadrilions of such elements in your head and they work in parallel (see also von Neumann Bottleneck why it is important that every processor has all memory available locally).
The more parallel machine you have, the more undeterministically it behaves. You cannot predict the outcome computed by ultimate intuition therefore, especially after it constantly self-learning by environment to solve unformalized problems from the environment. This makes it also impossible to grasp the algorithm of the brain that was used to derive the outcome. It reminds the quantum computation/coherence/uncertainty/miracles: the elementary you go, the more miraqulous your particles become (Penroze even foolishly decided that NN intuition miracles from Quantum Computation, which can also instantly reduce immense search spaces into a single optimal solution). You, thus may tell that computers are predictable while people are creative (unpredictable). Yet, that is another story. The point is that classical machines can imitate human mind but you need a lot of self-learning connecitons to diffuse your computation. Classical computers (especially supercomputers) can emulate neural networs but they still do not have enough processors to emulate the whole brain (the intellect is growing with the number of synapses and neurons, you know). The classical supercomputers are not just efficient enough to emulate the NN models. Special neural computers are 1000 times more efficient. Yet, even those machines still cannot simulate one human brain. Might be you can simulate a cockroach. It will not add numbers very well but it will bypass obstacles much more efficiently programer-hardwired logic. The latter is likely to never terminate because it will never be able to consider all situations (like blue brain in the chess contest). Intuition by its miracle just picks one, seeming the best and does not consider individual solutions, not even in the background. Yet, programmer will be able to approach the cocroach creativity developing more and more complex algorithm, likewise Deep Blue is looking more intellectual with more CPUs, more brute force and more complex algorithm. The classical approach approaches the intuition model as it becomes more elaborated and difficult to grasp. In the end, programmer may end up with the program, identical to what cocroach have learned naturally without hassle and this program will be a hell. You better start with NN right away.
The only problem why this was not done so far is that it is still costly to produce quadrillion of transistors/synapses. Nevertheless, the real brains are specialized NN hardware, after all and there is nothing in theory that can forbid miracles of artificial intelligence.
A bit messy but I think that the idea is clear. We need an efficient computer to emulate a neural network and thus we'll get an intuition. All the (classical) computers we have around are intended for execuing statements like "if input = this value then set that register/output that value, proceed to next statement". This way you program stupid, non-creative slaves, uncapable to learn and react immediatly finding more or less optimal solutions in the sea yet difficiency of required information. On the other hand, deterministic predictibility of the slaves is more efficient for routine, easy to formalize tasks like adding two numbers where intuition machine is overkill and may simply refuse to follow your commands being in bad mood, you know. You'll have to negotiate with AI.