Driving can be more difficult because the conditions make the vehicle the horse pulls vary greatly in its difficulty of pulling it. You can have only so heavy a rider, where the steepness of a hill and the depth of mud can increase to infinity if the conditions are bad enough. It can make a horse that can pull a light vehicle down the road with no trouble at all unable to pull the vehicle at all if the mud is deep enough and the hill steep enough. He has to pull himself and the weight of his load. The horse pulls according to his determination to work.
Some drivers do increase traction by using borium on the toes and heels of the horse's shoes.
Those who compete in combined driving might add studs for traction on slippery footing. Downhill, the driving horse must cope with the weight of the vehicle pushing him. The pressure comes from behind, and he has to hold a load that might be equal to his own weight. A horse is basically a claustrophobic creature. Some carriages and wagons do have brakes, so the driver can help the horses to handle vehicle and cargo.