This all depends on weather you can take the summer heat in the western states. If the answer is yes then I’d recommend driving Route 66 anytime between May and September. Coming from the UK I found the temperatures very hot but more than welcome! However, if potential temperatures of 120 degrees Fahrenheit scare you off then you might prefer to avoid July and August and instead focus on the spring/autumn months. It’s worth noting that most events/festivals occur during the summer months.
During the winter you could potentially come up against some severe snow and icy roads. Also, some attractions are either closed or operating on reduced hours during the off-peak season.