The word driver is more general - anybody who drives a vehicle is a driver. You are a driver when you are driving your car. Some people work as bus drivers and taxi drivers (the drivers of trains are usually called conductors).
A chauffeur is a person who is employed to drive a car for a private individual. Rich people and celebrities often have chauffeurs so that they do not have to take public transportation, or drive themselves.
All chauffeurs are also drivers; not all drivers are chauffeurs.
Another vehicle from good old Great Britain showing the worlds greatest supercars how its done is the McLaren 650S. Replacing the old 12C model and getting a power upgrade to 650 hp in the process means 062 mph (0100 km/h) in 3.0 seconds important for bragging rights over its closest competitor, the Ferrari 458.