Not to burst any bubbles, or cast aspersions on any of the Woolly-bear festivals that have sprung up (one in Vermilion, Ohio - the "Woollybear Festival" - began in 1973), but there is no credibility to the caterpillars' role as weather forecaster. The widths of the colored bands are unrelated to future weather. Their size is much more aligned with their age. Caterpillars grow through various stages, each termed an instar. Five instars/stages is a common number of growth phases for most of our caterpillars. As they grow, and shed their "skin" between instars, caterpillars often change appearance markedly. In the case of the Woolly-bear, the black bands widen with age, and thus older caterpillars are darker. Dark old specimens might lead one to believe a brutal winter is on tap.
For reasons unknown, Woolly-bear caterpillars are inveterate wanderers in fall and early winter. It doesn't seem to make much sense, as these caterpillars can eat an enormous variety of very common plant life, so there would seem to be little reason to risk the numerous threats that come with roaming far and wide. But nomads they are, and they know better than we what they are supposed to be doing.