This question is very good. Let me make a few points.
- First, it is important to say that no one version of English is considered more correct than the other. Both versions are accepted. This is an important point to keep in mind.
- Second, there are variations. However, the greatest variation is in spelling. Reading a little of both literatures will show you that there are slight variations such as in "theater" vs. "theatre."
- There are also grammatical variations and the nuances are slight. For example, in British English, the perfect tense is used to describe actions that just took place. For example, the British would say: "I have misplaced my book" Americans would simply say, "I misplaced my book."
- Here is another example with the use of possessions. Americans would say,"Do you have a book?" The British would ask, "Have you got a book?" They use the word, "got."
There are many other nuances and you must learn them on a case by case basis. In the end, both are correct.