What causes the brake lights not to work on a 1998 Chevy Lumina when the turn signal lights do work?

Your turn signal switch is warn out or broken. Try moving your turn signal forward and back gently while holding the brake on and see if your brake lights come on or flicker. The plastic clips that hold the turn signal switch together brake off or just wear out and because the brake lights run through this switch they don't work.

The pin switch for your brakes are under the pedal arm. Try depressing it by hand to see if they come on as it is adjustable (threads controlling degree of compression on button) if that doesn't work jump it with a jumper wire to bypass and see if they come on. If they do-fix switch, if not check the rest of circuit.