Q:

What are the differences between the different classes of the same car?

Some cars like the Subaru BRZ are available for purchase for multiple classes (drift, off-road, and race on this case). Because of this, you can the same car for different classes/disciplines.
What are the differences between the multiple classes of the same car? (let's say for example, the off-road and race versions of the BRZ)

A:

They affect the stats and handling of the car and also change the appearance of the vehicle itself. You can read more about the class breakdowns from this webpage but I'll give a brief summary here:

The Race class puts your stats in a pretty well-rounded field, decent speed, decent handling. This is obviously going to be the best for city-based races
The Off-Road makes your car have better traction on dirt, this is crucial for races over rugged terrain. Your vehicle isn't as fast to other cars on asphalt, but it seems to be a bit more durable too.
The Drag class. Straight-line speed. Nothing more, nothing less. Required for Drag races, of course, but is tolerable some other races. Keep it on asphalt or you'll be in dead last
The Drift class makes your car much easier to drift, and drifting can be held for longer a duration. You can't easily beat other cars in a race but you'll definitely be the flashiest driver there. Requirement for Drift races, but good on races with tight turns
The Runner class makes your car more agile and durable. With this class you'll be able to take tight turns and slip through narrow gaps more easily than you would be able to with other cars. You also get a boost to your durability and seem to manage car collisions better from my experience. A surprisingly well-rounded class, you can use this car and win just about anything

I'm not giving hard stats on any of these because that's more influenced by your car and mods installed rather than your class. A Drift class McClaren will still outpace a rinky-dink Drag class VW Beetle. Nevertheless, you will see the differences in your car's handling after driving cars set to each class. Throughout the game you are obligated to drive cars from each class so you'll surely be exposed to all of the driving types the game has to offer. 

