This is a very good question, and the answer could fill hundreds of pages - and, in fact, the answer already DOES fill hundreds of pages.
The short answer is that the figures you are citing do not agree with apparent reality because the commonly quoted figures are wrong :-). Read on ...
Much is available on the internet on this subject and the quality is, as ever, widely variable. There is also a lot of parroting of "facts" between sites and figures like those in Wikipedia seem common enough BUT there are some very reasoned arguments which seem to suggest that the Wikipedia figure is extremely wrong and underestimates the figure very substantially.
It's important to note that the eye acts as a contrast detector rather than an absolute level detector (such as a digital camera sensor uses) so comparisons need care.
With irising, chemical adaptation and every other trick it can pull it seems that the absolute dynamic range of the whole eye system is well over 20 stops. As each stop is a factor of 2, that's 2^20 or about "well over 1,000,000:1". At the top end, the sun is too bright!!!. At the bottom end the dark adapted eye can detect a single photon. A D3S (better performance than a D4) may have trouble with that. (Note that that is not EVERY photon - when you get down to the few photons per second level a lot of them will hit non-sensor areas and not be detected. But when one DOES strike a sensitive retina area it will produce a signal that can be recorded.)
But, I digress :-). An extremely good (it seems) page that discusses eye dynamic range and more is
Paragraph headings are worth noting:
Notes on the Resolution of the Human Eye
Visual Acuity and Resolving Detail on Prints
How many megapixels equivalent does the eye have?
The Sensitivity of the Human Eye (ISO Equivalent)
The Dynamic Range of the Eye
The Focal Length of the Eye
The writer argues that the dynamic range of the eye without changing sensitivity by adaptation or irising is about 1,000,000:1 in low light conditions. That is, as great as the "well over" lower limit mentioned above. Then he justifies this claim as copied below. This sounds fairly convincing at first glance. There may be flaws in the argument, but it seems OK, and this does not mean that it applies in all light levels.
Here is a simple experiment you can do. Go out with a star chart on a clear night with a full moon. Wait a few minutes for your eyes to adjust. Now find the faintest stars you can detect when the you can see the full moon in your field of view. Try and limit the moon and stars to within about 45 degrees of straight up (the zenith).
If you have clear skies away from city lights, you will probably be able to see magnitude 3 stars.
The full moon has a stellar magnitude of -12.5.
If you can see magnitude 2.5 stars, the magnitude range you are seeing is 15.
Every 5 magnitudes is a factor of 100, so 15 is 100 * 100 * 100 = 1,000,000.
Thus, the dynamic range in this relatively low light condition is about 1 million to one, perhaps higher!
But, here's a suggestion from me for an experiment at normal daylight light levels.
Find a scene that has a good mixture of dark areas and very bright areas - ideally with some dark areas as isolated islands near islands of brightness. An example may be sunlight shining through trees into a heavily shaded area - a few cavelets or deeply shaded areas will help.
Allow your eyes to adapt to the general lighting level - do not stare at the bright spots near where the sun is shining through and do not focus on any especially dark areas.
Note how well you can see detail in the darkest of dark areas - at what level of darkness does is fade to black.
Try the same with bright areas - as you look toward the sun there will be a place where details washes out and you cannot reasonably see more.
Cast your eyes to and fro across the scene between dark and light to try to stop your adaptation mechanism changing f-stop on you.
Now, take photos of the scene. Expose "correctly" and then so the darkest areas that you could see can be seen in the photo and then so that the brightest highlights you could distinguish are not washed out.
If you have the equipment, take an HDR photo with maximum f-stop variation between photos. (My Sony A77 allows 5ev steps.)
My experience is that my eye can always see a wider brightness range than my camera (Minolta 7Hi, A200, 5D, 7D, A700, A77, other)
On maximum HDR image (10 ev range between centers) my eye can see as well as or better than the camera.
The area where this does not APPEAR to be so is in extremely low light when I may need to allow the eye to integrate (which it does for up to about 4 seconds!) whereas I can look at a low light photo and see the image immediately. The fact that I may have needed a 10 second exposure is then irrelevant for viewing.
Other variably good stuff: