Noise gates sense the level of the signal input. When it goes below a certain level, it shuts off, keeping the signal totally quiet. Usually you set this level with a knob. This also means your sustained notes could get cut off when it goes below that level.
The hum debugger has no knobs, it does not ever shut off. It just somehow removes the hum. Everyone's waiting for it to come out so we can open and analyse what's going on in there. There's speculation that it's some intelligent digital circuitry that cancels 60 cycle hum.