It's not about whether the atmosphere does or does not spin with the Earth. It's about inertia. An object will move with a constant velocity until it is accelerated.
Here's a simple experiment to demonstrate this. Hold your keys or some heavyish object in your hand. Walk forward in a straight line and do not slow down or speed up. Now, throw your keys straight up in the air and continue walking with the same speed. Your keys should fall right back into your hand. This works on a moving train as well. The next time you get on a train, wait until you're moving very fast, at a constant speed, and in a straight line. Now, toss an object some distance in the air, and it will fall right back into your hand. The speed of the train does not matter, the object will always fall right back down.
The exact same principle can be applied to airplanes flying east or west. The speed of earth's rotation will not affect the plane's flight time at all unless the plane moves north or south. If you have a question or I haven't explained this clearly enough, please let me know.