The biggest advantage
you'll get from a constant aperture lens is the ability to shoot in lower light even when you're zoomed in.
The biggest disadvantage
is price and weight. The price will go up significantly for a constant aperture lens. Weight will also increase as it takes more glass to give you the constant aperture.
For example many consumer lenses such as the one you noted: 17-85 f/4-5.6 is able to shoot at f/4 at its widest angle, but the aperture closes down to f/5.6 at 85mm thus requiring more light to keep the same shutter speed.
If you buy something like the 17-55 f/2.8 (great lens by the way
) Then you get the ability to shoot at f/2.8 at 17mm all the way through to 55mm. Many of Canon's best zoom lenses have an f/2.8 constant aperture, 24-70 f2.8L, 70-200 f/2.8L etc.
Some refer to "normal" lens as being a 50mm prime (non-zooming) lens.