This button allows a greater amount of differentiation in rotational speed between the front and rear tires. ASR takes its cue from the front wheels; how fast they're turning is approximately how fast the rears should be. However, when tires become clumped with snow, their rotational speed varies to such a degree that the ASR will actually keep you from moving. Same thing when they have chains on, if the chains load up with snow.
I've actually had this happen in a snowstorm; hitting the button allowed enough wheelspin to get me going again. In my case, the rear end just stepped out to one side, and the car allowed a sort of power slide...very controlled.
If it weren't for this feature, I'd have replaced the button with a defeat switch long ago. But then again, I know how to disable ASR when I want to
Hope this helps