Felt I had to jump in here with my 2c. worth......
I am British, I was educated at an expensive school in Cheltenham in Gloucestershire so I tend to speak that way. Although I have lived in The States for almost 20 years, most of it in The Deep South, I am married to an Englishman and have lost none of my English accent.
Now, I KNOW the "expensive," word will jump out at everyone in the paragraph above. The reason I put that in, (before I am called all sorts of nasty names) is because we definitely had a different accent to the locals, or children from other schools. We developed a more "far back" English accent, so called because we tend to speak from the back of our throats rather than the front of our mouths as people from the north of England do for instance. This tendancy seems to be going out of fashion with the younger generation, just listen to the way Prince William and Prince Harry speak compared to their father.
This accent, when I was at school, was the favoured BBC accent which seems to have almost disappeared now in the English media. Strangely enough it is very favoured over here, especially when actors are playing English butlers! The Americans in general (not the American media) are not very discerning when it comes to a British accent. They do not seem to be able to tell whether I come from England, Wales, Scotland, or Ireland, or even Australia, or South Africa. They also think variously that I speak like Paul McCartney or Mrs Thatcher!
The Southern American drawl, (not South American) is quite different from the accent California Jim will have, but the majority of English people would not be able to tell the difference. We recognize Americans by their monotone or when they say "bedder" or "innerrupted" or do not pronounce our unique vowel sound as in "what" (which Gwyneth Paltrow and Madonna have gone to great pains to perfect).
As to which is best, the answer is neither. From my experience to avoid misunderstanding it is best to adapt ones accent according to the majority. When I first came to the States I remember going to buy a pen as a gift for someone, I was directed by the sales person to the kitchen department to buy a pan. All the different American and English accents give the language variety and colour, The Americans are exceptionally good at making up words and descriptive phrases which sum up exactly what you want to say. "bummer" is a good one I use a lot.