Grip.

[crazy legs]

Does surface area affect grip?

[heathfield]

Yes, and no. Friction depends upon weight or pressure. For a given weight, by increasing the area, the 'weight per unit of area' is decreased in direct proportion, and the total friction remains the same.
If you increase the area, but maintain the 'weight per unit area', (grip harder) then the friction will be increased.

[kwicky]

Yes all surfaces have a coefficient of friction.

[chakka35]

Yes. Smooth tyres on a dry road give more grip than treaded tyres because there is more tyre area in contact with the road. We need treads to squeeze out water on wet roads, otherwise the water would build up as a wedge in front of the tyre and cause disastrous aquaplaning.