Wikipedia has a great summary: 

https://en.wikipedia.org/wiki/Comparison_of_American_and_British_English

The differences are actually very small. The Wikipedia article notes that within the U.K. itself, "dialects, word use and accents vary not only between England, Northern Ireland, Scotland and Wales, but also within them." U.S. English is probably no farther from standard British English than Scottish English is.

The reason why the difference has become exaggerated is related to history and cultural identity. First, in 1828, the American lexicographer Noah Webster, for whatever reason--national pride or a good marketing opportunity--published "An American Dictionary of the English Language," in which he changed the spelling of some words--colour to color, traveller to traveler, centre to center, and so on. This reinforced the idea of an "American English" as being different and separate from British English, but the same throughout the U.S.

The second reason is that printing and publishing flourished in the the British colonies, and we evolved a tradition in which British books were normally published and distributed in the U.S. by U.S. publishers--who edited the text to conform to U.S. expectations. ("Nappies" to "diapers," "elevators" to "lifts," and so forth). Book titles were often changed completely. Notorious, Harry Potter and the Philosopher's Stone was changed Harry Potter and the Sorcerer's Stone.

Some of the differences are really silly. In the U.S. we put periods and commas inside quotation marks, and in the U.K. they put them outside. 

In the U.S. we use double quotation marks for the outermost quotation. I was taught in the U.K. the custom is to use single quotation marks for the outermost quotation, but this seems to be changing.

"This is how we punctuate quotations it in the United States," the American wrote to his British friend.

'Well, that is a very bad way indeed', she wrote back.

British people say; I'm going to hospital. 

American people say; I'm going to the hospital. 

British people say; I have maths class, or, I hate maths.

Americans say; I have math class, or, I hate math.

British people do sums. Americans do math problems or arithmetic. 

These are just a few things I know of that are different. 

Hi some basic words have differences

1. British (colour) vs          American (color ---with a sress on 'r')

2. Bank cheque  vs              Bank Check  

3. lift                                      elevator

4. cinema                            movies

5. petrol station                    gas station