Bias ply tires are created by stacking layers of nylon, rayon or some other fabric material to create the body of the tire. This stacking of layers makes them somewhat stiff when sufficient plys are used for over the road use. This stiffness in the casing can cause the tire to distort under hard use, which can compromise traction. This is why a bias ply tire has a very square shoulder to it, to prevent the tire from rolling over. Rolling over on the edge causes distortion problems. Additionall, all these layers of belts make the tire heavy and allow them to retain heat, which will shorten the lifespan of the tire as well. Over the road use of bias ply tends to have the car wander as the tires follow ruts and grooves in the road. Additionally, the multiple, heavy layers tend to flat spot when parked overnight and can cause a sometimes severe thumping when driven the next day until the tires heat up and the flat spot is worked out. The wider the tire, the more pronouced these issues become.
Radials use steel belts that wrap around the carcass of the tire in fewer numbers than belted tires, so that gives it considerably more flexability. It also allows stronger construction with a lighter weight so they are more responsive to irregularties in the road. This also minimizes distortion under hard use which allows greater contact area, which enhances traction. The lighter weight combined with the rounded shoulder allows a radial to roll over road irregularities rather than be directed by them.
In competition situations, I've noticed that both grip almost equally well, to a limit. One big difference between them is that a bias ply will tell you it is going away. The noise and gradual loss of traction allow the driver to hear and feel the tractive situation and adjust for it. Recovery is easily obtained and can be done on the fly. Radials, on the other hand, are like driving on ice once they let go. You will need to almost completely back out of it and allow the tire to regain grip before hammering it again. Bias ply tell you when your pushing the envelope, radials don't, they just cut loose. However, once the driver adjusts to these difference, the radials will be faster more consistently. This is why you still see so many bias ply drag tires out there, but not very many biased handling applications.
For all around street use, the radial is better all the way around. They are lighter, more responive, just as fast, grip as well, and last longer. To throw a guess out, I'd say 10-25% better than a bias ply tire run through the same situations.
