I had the same problem with mine when I bought a used 97 A8. Made the noise when I turned it on, but did not move. I took it apart and found that there is a small motor which drives a cable (like a speedometer cable on a motorcycle), which in turn connects up to the headrest. When the motor is activated, the cable drives a gear which moves the headrest up and down. Pretty simple. The problem I found was that the hard black plastic outer sleeve of the cable was actually LONGER than the metal cable contained within. So, when I tried to lower the headrest, the actual metal cable inside the black outer sleeve was just spinning away, not making contact with any of the gears up at the headrest. The black outer cable has a coupling on each end, so I removed the coupling, shortened the black casing about 1 cm, and then pinched the coupling back on. I then reattached everything and the black outer casing now did not stop the inner cable from making contact with the gears. The headrest works great, and I saved a few bucks over ordering a new assembly. It is a pain getting to all of this inside of the seat, but everything in this car goes back together a lot easier than most cars I have worked on.

Hope this helps!