It's a mixture of bhp and angle/speed when drifting.
Every tire will lose grip at one point - hard ones easier with lower speeds/momentum, soft ones with more speed/momentum.
At lower speeds (usually couples with lower bhp car), you need less grip in order to drift. Therefore harder, more narrow tires
At higher speeds you need softer and wider tires to be able to control the car. The bhp and torque for a competition drift car will be sufficient for the wheels to spin. Also, professional cars are equiped with locked differentials, so this help a lot too.
I have no idea what the difference between road, sports and race tires are in the game, but my guess would be this rule applies to all tire classes.