My roommate and I have been arguing about this and I need help from someone who can explain it.

If it were possible, and you could drive the whole way around the the world. Would the spin of the earth effect your drive time. Basically depending on which way you drove no matter how small would there be a difference in time. Of course this is saying everything else is constant.

I don't think so. Because if you touch the ground then you are spinning with the Earth (contrarily than if you take a plane, there will be some difference) so that going to East from West would take as much time than going from East to West.

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Isaac
If the problem is too hard just let the Universe solve it.

so even though one way you would be moving into the spin and the other with the spin it would still be the same?

Yes, that's it. (at least according to Newtonian Mechanics I believe)

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Isaac
If the problem is too hard just let the Universe solve it.

Because the car is already on the earth, it already has the earth's surface velocity. That is why we experience no movement though the earth has a very high surface velocity. All motion is in this frame so there would be no difference.

Even if you were to use a copter and fly vertically up and stay there, you won't reach new york just because the earth is spinning below you!

The copter already has the same horz vel as the surface, and when you fly straight up, there is no force in the horz direction and by Newton's law, it continues to move as before horizontally.
Else air travel would have been so much faster!

However if you were to count time by the sun, you would see one sunrise less!