If I change the aperture for a particular photo, why don't I see a change in the brightness of the image in the viewfinder?
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).
If your camera has an electronic viewfinder (DSLR or mirrorless) the brightness of the viewfinder is set independently of the brightness as exposed. Often these cameras have a setting buried in a menu somewhere to adjust to the screen's brightness to the actual exposure that the camera anticipates using.