When an airplane starts in any direction, its velocity with respect to any reference frame automatically gets the contribution from the moving Earth's surface.
Equivalently, you may look at the whole situation from the Earth surface's viewpoint and then the Earth's rotation is invisible and can't influence the speed and timing of flights, because of the principle of relativity. In this idealized description, there's no difference. This conclusion would be right when we neglected the atmosphere, mostly for e.g. the rockets that spend most of the time outside the atmosphere.
However, there exists the atmosphere and it has winds - whose average speed ultimately depends on the Earth's spinning as well but the dependence is indirect. In moderate zones, the westerlies dominate – winds from the West
and because the airplane flies in the atmosphere and wants to reach a particular speed relatively to the air mass, it's clear that the speed of westerlies helps you when to speed you up when you fly from the West, and slows you down when you fly to the West. That's why flights from America to Europe (or from Sydney to California) are about 1 hour faster than the opposite flights.