This dispute has died down over the years as the skiers have stolen ideas from snowboarders (terrain park riding, twin tips) and older people have started to take up snowboarding as they realized how much easier it is on the legs compared to skiing, especially in deep snow.
Being someone who alternates between the two sports, I might be in a good position to make unbiased comparisons.
I have come to the conclusion that there is only one difference between skis and snowboards that is worth talking about: snowboards have only one short edge, whereas skis have two long ones. This results in snowboards having less edge traction than skis, which means it is harder to stay in control and the act of staying in control usually involves more sideslipping. Sideslipping is what does the real damage to the snow; the effects of a snowboard's wider track are pretty negligable. Also, since you can't turn as fast, it is more difficult to navigate moguls that have been created by skiers, which usually means you end up "destroying" them if you try. Moguls that have been tracked over by snowboarders usually consist of widely spaced piles of snow sitting on a sheet of ice.
Do I think that snowboarders should be banned from ski areas for these reasons? Definitely not, although I would like to see each ski area designating several runs as snowboarder-free mogul runs. Where I live, this would barely be necessary as the ski areas around here get snowstorms often enough that any wrecked moguls or ice will be covered up by new snow fairly quickly. In powder, edge length is less of a factor in producing traction, so snowboarding is more fun and less damaging in powder. For this reason, a lot of the snowboarders around here either go off into the backcountry or stay in the park during days when there's no powder in the ski area. I usually just go skiing instead.