The specific reasons vary depending on the type of vehicle. Two distinct examples immediately come to mind.

Example #1. The Tractor.

This John Deere tractor makes use of large diameter tires on the rear driving axle and small tires in the front for steering.

Tractors use rear wheel drive because they are designed to tow trailers or pull lawn mowing/tilling equipment. Dynamically speaking, adding a trailor to a vehicle increases the down force on the rear wheels while having no effect on the front wheels. Similarly, a snow plow add-on will, in use, potentially decrease the down force on the front tires while having minimal effect on the rear tires.

Big tires on the driven rear axle allow for more grip and reduce the chances of the vehicle digging itself into a rut or getting stopped by a low barrier.

Big tires mean more surface area resulting in less soil compaction, which is good for farmers tilling fields.

OK got it! Big tires are good for tractors, so why the small tires on the front?

*Smaller tires are a lot cheaper to buy and design around, and are easier to replace.

*Smaller front wheels allow the vehicle to make tighter turns. A very tall front tire would hit the engine block if turned too sharply. For those massive tractors that require all wheel drive (and so all big tires), a different steering mechanism must be used. These designs require a hinge in the center of the chassis.