They cut those parts out because it's a video for mostly metal stations. Typically, a majority of rockers won't want to sit through a bunch of trip hop beats to get to the "juicy" rock stuff. So they figured that the quicker they get to the guitars and upbeat tempo, then the more likely most rockers who are not yet familiar with Katatonia will want to pay attention and watch/listen.
As for the vdeo itself, it's not my favorite. It's better than "Day And Then The Shade" but I seem to favor "My Twin" as my top Katatonia music video.
Jack