Route 66 was originally designed to follow the southern route across the western half of the USA in order to avoid the high mountain passes that were typically closed during winter months. Even so, the old route encountered winter snows and bad weather along certain parts. Illinois, Missouri, Oklahoma and Texas can sometimes get winter storms with high winds and blowing snow that make driving hazardous. The area around Flagstaff and Williams Arizona is near 6,000 feet (2,000 meters); so it tends to get a bit of snow during the winter months.
Most of the way, you will be driving on modern interstate highways like route 40. Those highways are typically well maintained and kept clear at all times of the year. You will only be getting off of those modern highways to search for the remaining segments of old route 66. They might not be maintained as well as the interstate.