I would say that the difference is, in part, a matter of scope. American History tends to focus on the history of the United States, beginning largely with the colonial era and tracking forwards towards the present. It's largely a national history. World History, on the other hand, is far more encompassing, both geographically as well as temporally.
The thing is, the history of the world is far larger than any one place—it encompasses the full breadth of human history, starting with the earliest cultures to emerge in the Ancient World—cultures like Egypt, Sumeria, the Indus River Valley civilization in India—before tracking forwards. It covers various different geographical regions as they unfold: China, India, the Mediterranean World, the Americas, Western Europe, and so on, stretching across different time periods on both sides of the Common Era.