The risks of driving on a tire with sidewall damage include the tire going flat; damage to the car, particularly the rim; and the tire blowing out. These risks are much more likely with damage to the sidewall of the tire than with damage to the tread.
Tire sidewalls measure about 1/4-inch, which is significantly thinner than the treads of the wire. As such, they are harder to repair effectively. Even with a repair that appears to hold, there is still a reasonable chance of the tire failing.
The most dangerous risk is a blowout, when the tire bursts, resulting in the rapid loss of air pressure. Blowouts can cause the vehicle to suddenly veer to the side, a problem often made worse by the reactions of surprised drivers.