There is actually a fairly fundamental difference in the design. The diaphragm (the part that forms the aperture) in almost any lens is somewhere around the middle of the lens. In a fixed aperture zoom, only the elements behind the diaphragm move around to do the zooming. In a variable aperture zoom, elements both behind and ahead of the aperture move around to do the zooming.

At least in the usual case, the diameter of the aperture does not change as you zoom. This is fairly easy to verify -- take pictures at different zoom ratios and maximum aperture with some out of focus highlights. At least with your typical zoom lens, the out of focus highlights will remain round at all focal lengths, indicating that the aperture is remaining wide open (where it's round). Stop down the lens a few stops, and you'll start to see the shape from the aperture blades closing (though lenses with lots of blades, especially rounded ones, will retain nearly-round looking highlights somewhat more than others).

When/if the elements in front of the aperture move around during zooming, you're changing the (effective) focal length of that part of the lens. You're then transmitting light through a fixed-diameter aperture, meaning the (effective) f/stop changes. Since it's only affected by the change in effective focal length of the elements in front of the diaphragm, the change doesn't (usually) correlate exactly to the change in overall effective focal length -- moving the elements behind the diaphragm changes the effective focal length without changing the effective aperture (e.g., my 28-135 has nearly a 5:1 zoom range, but the aperture only changes from f/4.0 to f/4.5).