It is probably better to say that Britian speaks English and America speaks American. American is based on English, and is mostly the same, but there are significant differences, especially in the spoken language:
1. Accent. A "proper" English accent like what is used by TV Newscasters is easy for each other to understand. However, slang like cockney in England and a heavy southern American accent may be quite difficult for each other to understand.
2. There are many words that have different meanings. Examples: Bonnet (in America, it is a type of hat. In England, it is the hood of a car); Boot (in America, it is a type of shoe, in England it is the "trunk" of the car); napkin (in America, it is on the kitchen table used to wipe face and hands as you eat, in England it is something women use once a month); There are many other examples of this.
Each English speaking country has developed it's own vocabulary and accent - British English is different than American which is different from Australian which is different from India which is different from Hong Kong English. Most consider British English to be the standard formal English.