What do brains do when processing information, that Von Neumann machines cannot (yet?) imitate? 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 &quot;how does the human brain differ from a computer?&quot;.  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 &quot;evoke&quot; or &quot;activate to varying degrees&quot; 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.
EDIT:
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 &quot;spillover&quot; 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 (&quot;self&quot;) that arguably constitute a concept.

References:
Fletcher, P. C., Shallice, T., Frith, C. D., Frackowiak, R. S. J., &amp; 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., &amp; Brody, C. D. (2005). Flexible control of mutual inhibition: a neural model of two-interval discrimination. Science, 307(5712), 1121-1124.