Why don't I see a difference in the brightness through my viewfinder when I change the aperture?

You will not see a change of brightness through the viewfinder as the aperture is only stopped down when the photo is taken. This allows the auto focus sensors to focus accurately as many cannot focus below certain apertures. It also provides more light for you to compose your shot with.

Many DSLRs have a "Depth-of-Field preview" button that stops the aperture down to what it will be when the photo is taken. You will notice a difference in the light and depth of field when this button is pressed but only if the chosen aperture is both smaller than the maximum of your lens and what your focus screen lets in. This can be anywhere between F/2 (measured) and F/3.8 (reported).