I opposed to what AMD said. I also read about this when I just ordered copper spacer. I had a chance to wonder whether I have to use it when it arrived.
I finally used it for comparison.
My system Duron700@1037 is running 53 full load, and 49 idle (I'm in the tropics where the climate is about 32-33 C). I couldn't achieve this without the spacer. It does help some heat transfer to the heatsink.
I've tried it without the spacer, and the temperature is about 3-4 C hotter.
About the NON_CONDUCTIVE thermal grease, yes you should use the thermal grease over all the bottom of the spacer. This is a must, and the warning is also explained clearly with the instruction paper. Or else, you are risking of shorting the CPU. With non-conductive grease, you are much safer.
It is even better to apply to both sides of the spacer , as I did, because it does help lower the temperature.