Same thing.

They mean exactly the same thing. In the United States, we use both terms frequently and almost interchangeably. "World War II" is the most common way to refer to it. It is neither formal nor informal. You can use it everywhere. For example, it is used in history textbooks and officially. "The Second World War" is very slightly more formal. For example, Winston Churchill wrote a six-volume history, and the title he chose was "The Second World War." It is always written as "World War II," never "World War 2." It is read aloud as you would expect, "World War Two." It is sometimes abbreviated "WWII" but that IS informal.

Same thing