Different Floors

Is friction significantly different between the competition tiles and a regular classroom - hard floor - tile, and even carpet? If so, will it affect the driving performance during the autonomous period?

Not if you use encoders, but it will with wait commands.

I agree with @briancole, definitely use encoders. But if you don’t, then I would say the friction is similar to low-pile carpet. It may not affect your driving straight, but it probably would affect turns if you don’t use encoders.

I would say it is very noticeable, depending on the speed and weight of your robot. On a tile floor I can get my bot (turbo) to drift sideways over a foot in some turns, but that would only be an inch on the mats.

Yes, the pile of the carpet can negatively effect the robot driving performance. Especially if it is a heavier robot.

Get some plywood or underlayment to stick under the field to mimic a gym floor.

Even if you are using encoders it will make a big difference.
Even for going straight
Even if you are using PID, the tuning would be different