If you want a full and comprehensive answer, it's here: https://en.wikipedia.org/wiki/Comparison_of_American_and_British_English If you want a simple answer, it's this: The differences are NOT as great as non-native speakers believe. The British and the Americans ( and the Australians, Canadians, New Zealanders and everyone else) all speak the SAME language. There are some differences in vocabulary, but we are generally always aware of the vocabulary used elsewhere. For example, your textbook may tell you that Americans say 'garbage' while British people say 'rubbish'. But the reality is that Americans occasionally say 'rubbish' in some contexts and that the British often say 'garbage'. What's more, we ALL understand these words, so it doesn't matter which one you use. The grammar is almost identical, and - give or take a few differences in spelling - the written language is the same. It really doesn't matter whether you write 'center' or 'centre', 'colour' or 'color'. These differences are very minor. There are some differences in pronunciation - such as some vowel sounds and whether or not the 'r' is pronounced after a vowel - but again, these differences are relatively minor. Unless someone is speaking with a very strong regional accent, we can always understand each other. What's more, the differences are getting smaller all the time. British English is becoming more and more similar to American English. Who knows.. in a few years' time, there may be no difference at all.

There's almost no difference in the written language in newspaper stories or nonfiction books. There are some differences in small details like the spelling of a few words. The pronunciation is a little different, particularly in some of the vowel sounds--I'm not sure a beginner would even hear them. For example, if someone speaking "BBC English" says the word "short," to U.S. ears it sounds almost like the word "shoat." An American named Noah Webster published a dictionary in the 1800s. If it weren't for that, and the existence of national boundaries, we would just consider U.S. and British English to be "regional dialects." The differences do not impede understanding.

I don't notice any differences most of the time. Sometimes I see spelling differences: colour/color, travelling/traveling. Americans say, "Did you have breakfast yet?" sometimes, whereas I would always say, "Have you had breakfast yet?"