How does one come to answer this question?
Power can be thought of as energy per unit time. For example. 1 Watt = 1 Joule/sec.
Thus, a car with higher power means a car that is generating more Joules. So lets get into the mindset of Coservation of Energy:
PE + KE - Friction = Total Energy
In order for a car to move forward, the car relies on transferring its kinetic energy into Friction energy. A car with more power, will also have a higher Torque (Power = Torque * angular velocity). This higher torque means higher kinetic energy. Higher kinetic energy means more energy is being transferred into friction.
The car with higher power can benefit from wider tires as it will increase the surface area through which friction force is applied. This increased surface area means the increased friction force is spread out and does not wear the tires as fast either.