Not sure exactly what you are asking, but an give you some ideas about how the British accent evolved into the American one in North America over time.
When Britain established its first permanent colony in North America, many of the immigrants were ethnically English, and the ones that followed often were Scots-Irish indentured servants. Over the next 169 years before independence was declared, a flood of immigrants of many nationalities, including French, Germans, Swiss, Dutch and African slaves all arrived on the continent, bringing with them their own dialects and languages.
Over time, most of these groups intermixed and intermarried, and the ethnic lines between settlements blurred. Vocabulary was borrowed from language to language, and new, uniquely American terms emerged. By the 1750s and the French and Indian War, the British soldiers and officers who were sent the North America observed that "Colonial English" was quite different than what they spoke, in terms of accent and content.
The process of continual accent and vocabulary change has continued into the present day.
This is a fairly broad question and the answers you receive might reflect that. I would say that if we looked at the cosmetic differences between the manner of speech in both accents, pronunciation is probably one major difference I find. You might find more American accents dropping off letters in words, such as the “g” in words ending in the suffix “ –ing.” Of course, this is not to say that we don’t see the same thing in British accents in different parts of the United Kingdom, but I think it is a fair statement to make. I see the spelling differences as something that has persisted throughout, highlighting potential differences in speaking. For example, “color” in America and “colour” in British English is the same word, but with different spelling. Finally, I would suggest that accents play a vital role in the cultural stereotypes of each culture. The American accent, if there is one because regionalism in America is a challenge in defining “American accent,” is used to portray them as more brash and oblivious to delicacy and nuance. In contrast, the British accent, again challenging as a whole to totalize because of the divergence in speaking throughout the United Kingdom, is depicted as only nuanced base, and lacking any sort of intensity. A great view of the differences in accents and cultural representations would be from the film, “A Fish Called Wanda,” where Archie (John Cleese), the British banister, is chiding George (Kevin Kline), the American mercenary, through accents differences which are reflective of historical rifts between both sets of cultures. In both, the use of the American and British accents play a vital role in how “the other” is perceived.