The biggest effect in time difference is due to winds, whichever direction they happen to be blowing at the time, which is why sometimes flights between the same endpoints use different routes so as to avoid them. The atmosphere rotates along with the earth which is why e.g. the mushroom clouds from the two atomic bombs stayed above their targets. There are two effects which come into play here: Coriolis and centrifugal.
The Coriolis effect is most pronounced when flying from a pole to the equator. During that flight, the plane will need to build up the extra 1,000mph that the Earth is moving at the equator. Or, put another way, the plane propulsion will have to compensate for the Coriolis pseudo-forces.
The centrifugal force comes into play because the earth is curved, which means that the plane has to change its speed in a vertical way. However, at this point, the gravity is actually helping the plane because, due to the earth's rotation, the gravity is slightly less than it would be if earth didn't rotate. So the plane needs to produce less lift.
But there is absolutely no difference in flying West or East.