The difference is width. 8/9/10 speed cassettes has all the same width, so the more speed the tighter the sprockets are bunched together. The tighter the sprockets are spaced, the narrower the chain has to be. You may be able to use a 9-speed link on a 10-speed chain, but it may also snag catastrophically on the (front) derailer.
dabac is correct about the width of the chain/link. The more gears the tighter/less space the rear cassette has. You can use a higher number chain/link but not a smaller number. If you have say a 9 spd cassette you can use a 9,10 or 11 spd link. If you used a 8 spd link you would likely have wear and binding problems each time it went through the cassette.