The main reason that race cars are fitted with very stiff suspension is so that the downforce doesn't cause the race car to bottom out.
With softer suspension the car would bottom out due to the downforce of the race car.
Regular vehicles have softer suspension and springs to allow the vehicle to adapt to road irregularities much quicker so that it doesn't cause the vehicles tires to have a loss of contact with the road, and thus provide more mechanical grip.
The softer suspension on a regular vehicle gives the regular vehicle a smoother ride as well but race cars do not require as smooth of a ride as a regular car does so they're fitted with the stiff suspension to avoid the race car bottoming out.