They are both VERY big states. Please add a detail indicating from what city to what city. It makes a difference.
Definitely some snowy patches depending on your route. Be sure to review the free winter driving hazardous roads conditions course (free) before heading out. Will help you ensure safety!
Peedle makes a good point. Are you going from Brownsville to San Diego or from the panhandle to Eureka? Add a little more info please.
If you take I-40 there could be some snow in New Mexico or around Flagstaff AZ