Yes, the special design is the design/width/depth/position and design of the treads in the tire, that allow them to contain more ice and snow (or mud) before losing their stability, and also lets the tire get rid of the built up debris quickly via designing of the channels. In general, winter (or mud and snow) tires perform better in lower temperatures because the rubber is a bit softer, which allows the tire to grip any non-leveled terrain (such as ice buildup) better. Because they are softer and grip better, using them in the summer when the roads are at the optimal condition will cause them to wear down much quicker. So for the longest lifespan of your winter tires, only use them for winter and spring and maybe late fall, and switch to summer tires for the summer. Regarding your comment about having an accident where the tires played a part – certainly it could be a factor. But if you live in a zone where it’s Winter now, there are many more factors that are much more likely to have contributed to the accident than the build of your tires.