That is a very interesting question. I guess the short answer is: Clouds do move. They do not just appear to move because of the spinning of the earth. However, the direction that clouds and other air masses do move, is definitely affected by the spin of the earth. If you want to learn more about this effect, look up "coriolis effect" on google.com

Clouds (water vapor) are part of the whole always-moving atmosphere of the earth. We just notice the movement of the clouds because we can see them. The rest of the air is moving too. Wind is also a sign of this air movement. All this movement comes about because different parts of the earth's atmosphere are heated to different levels by the sun. The energy imbalance causes air to move.