In reverse order: first of November, there's about 11 hours of daylight (probably good cycling for 9.5 hours) a day. By the end of December, you're down to less than 8 hours of good cycling daylight.
Rain: October is our driest month (especially with the current drought!). Probability of precipitation is going up from 30%/day, on average, to 50% per day. November is when we transition from pop-up thunderstorms to powerful fronts coming in.
Temperature: Early November is great cycling weather, usually mid-40s to mid-60s. Late December, not so much: how do you like cycling in frost in the mornings, with highs usually between 40 and 50? It's tough commuting by bike then, since I end up having to bring home cold weather gear 2/3 of the time -- unless the temps are going down, then I wish I'd brought some more. Proper equipment helps a lot, but on some days you're going to be spending a lot of time changing clothes...
Are you thinking of camping or B&B/motels? Many of the campgrounds on the Natchez Trace will be closed, but a warm shower and toasty bed at the end of a long, cold day are great luxuries.
Edited to add: You probably won't have to worry about snow. South of Tennessee, the snow removal plan is typically "wait 'til it melts." And that plan only gets exercised every 3-4 years.