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I used to work for a major tire manufacturer.
Changing the diameter of the wheel effectively changes the gear ratio of the drive train. This applies to cars and bikes.
A larger diameter wheel gives more leverage, effectively "gearing down" the vehicle. A smaller diameter wheel gives more RPMs over the same distance, effectively "gearing up" the vehicle.
Bicycles have larger wheels to make them easier to pedal. If you've ever ridden a folding compact bike, you may have noticed the smaller tires make them harder to pedal or have larger gears to compensate.
Car tires have a smaller diameter to reduce material cost. Truck and SUV tires tend to be a little bigger to pad the final torque figure. Tractor tires are huge to maximize torque. Big offroad tires are mostly to give more ground clearance, the extra torque is just a bonus. The reason offroad tires aren't as big as tractor tires is because they still need to fit under a wheel well and because there are height restrictions on street legal vehicles and people occasionally drive a few miles on public roads in their 4x4.
Tread width improves cornering and acceleration performance, to a point, by giving a larger contact patch, which gives more friction, but reduces fuel efficiency and eventually reduces performance. Economy car tires aim for a good balance between braking performance and fuel efficiency. Sports cars aim for a tread width commensurate with the desired performance capabilities; wide enough to put power to the road and stay stable through the corners, but not any wider, so as to keep weight to a minimum.
Bike and motorcycle tires have a curved bottom to give traction while leaning, so this is factored into tread width also.
Tires are part (sometimes the only part) of the suspension of any vehicle. The air in the tire gives cushion. More air means more cushion. Higher tread wall means more room for air. The height of the tread wall is determined by giving enough to soften the ride, but not too much because tires are heavy and additional mass means lower fuel efficiency and reduced performance. Tires are considered "unsprung weight" because they're not supported by the springs of the suspension. When calculating performance, one pound of unsprung weight is equivalent to approximately three pounds of sprung weight, so optimization is important in that area.
Weight capacity, wear rate, and shear strength are determined mostly by the profile design, material composition, and manufacturing process. Diameter and width aren't (usually) adjusted to suit those parameters.