detects "Closeness" (for want of a better word)

The first allows the screen to rotate when you move it in different directions and also detects any change in position, like if you were using it as a steering wheel in a car app.
The accelerometer measure directional movement, like swinging your arm in a bowling app.

Proximity sensor shuts off the screen when you put the phone to your ear. No need for a bright screen unless you can actually see out of your ear, which I assume you can't
The ILS controls the screen brightness based on the ambient light in a location.

Yes, they are complementary to each other and combine to provide added functionality to the iPhone.