Edited 2011-02-19 for better wording. Step 3c is showing TWO stops. Single stop tip added.
Considering light capture, f/2.8 compared to f/4.0 is the same as the difference in switching from ISO 200 to ISO 400.
For using very short focal lengths, the shorter DOF created by a larger aperture may be one difference – if you go REALLY close to your subject… not in other cases. The difference is a lot more prominent at longer focal lengths, and increases with the focal length.
For a Tokina 11-16/2.8 set at f/2.8, DOF really doesn’t exist beyond 2 meter (6′) focusing distance at f=11mm and beyond 4 meter at f=16mm, because you hit hyperfocal (check here).
So for short focals the practical difference is more light into the viewfinder.
Now to see THE AMOUNT of the difference:
1a) Pick up your camera,
1b) set it to “A” (Av for Canon) mode,
1c) use your controls for setting aperture and turn it to the lowest possible number.
1d) That is the largest aperture your lens is capable of at that focal length (for a zoom).
2a) Now double that number and
2b) turn the dial to find the new number.
3a) Find the “preview” button on your camera and press it.
3b) Look in the viewfinder and release the button.
3c) Re-press the button a few times to get the feel of the difference.
The difference you see in the viewfinder at step 3c) is DOUBLE the difference you will see going from an f/4.0 lens to an f/2.8 lens.
If you know how to dial in a SINGLE stop (multiply by 1.414) in step 2a, then try that too.
The exact amount of light you see in the viewfinder will always depend on the scene before you…
the DIFFERENCE in the viewfinder with an f/2.8 compared to a f/4.0 lens will be exactly as the SINGLE stop tip above.