And: since when is it that different? Did Lincoln had an American accent?!
For instance, the "standard" (TV broadcaster) accent originated in areas settled largely by German and Scandinavian immigrants. The "New York" and "New Jersey" accents have elements of Irish and Italian and Yiddish accents, and New York accents were influential due to being common among entertainers through the middle of the 20th century.
And accents can evolve quickly. The New York accent you'll hear in films from the 1950s is very different from the one you'll hear today if you go deep enough into Queens to find strong accents.
That said, IMO moderate Scottish accents sound extremely similar to the "standard" American accent. I think it's actually that the cadences are very similar, which for me at least is what makes other strong British accents (other than RP) often difficult to understand.
Accents tend to shift based on contact with other linguistic groups and mixing from different accents that levels out pronunciation. You'll note, for example, that Canadian has more in common with the US accents than it does with British accents. I don't know the historical development of accents very well, but some US accents (I believe the southern one in particular) are closer to 16th and 17th century English than modern British accents. This isn't necessarily atypical--modern Icelandic is the closest in pronunciation to Old Norse, while modern Norwegian is closer in pronunciation to older varieties of Danish. Small, isolated communities are going to be able to better preserve pronunciation in a dialect.
It used to be even more pronounced, with neighbouring villages in England having distinct accents. Nowadays, i suspect that the differences are disappearing - although perhaps some will always remain.
Similar happened in France and French Quebec. Modern French is formalised Parisian french. While Canadian French is country French, according to a very posh friend, the most educated quebecois still sounds like a country hick the first time he hears them.
You got it backward. Modern French is bourgeois Parisian French. It was standardized more than a century after the French revolution (in the 1880's when public education became mandatory).
When Québec was colonized in the 1600's, only a handful of territories in France spoke French, mostly around Paris. This was the language of the court, the nobility and a minority of peasants. The rest of the country spoke various languages, such as Caló, Catalan, Corsican, Franco-Provençal, Ligurian, Occitan, Flemish, Luxembourgish, Alsatian, Breton, and Basque [1]. Furthermore, Parisian French was only one of the d'Oïl languages spoken at the time [2].
The colonists that settled and explored Québec were mostly from northern France, the region where an Oïl language was spoken. The language spoken in the colony was standardized to match the one of the royal administration (Parisian French).
Due to not being a French Territory when the revolution happened, Québec kept a fork of Royal French. Reading written french from the 1600's and 1700's out loud is striking, as it's almost identical the pronunciation of modern Québécois French. An example of this is the term "Moi", spelled Moé in the 1700's and pronounced Mo-a (Moi) by modern continental French and Mo-Hey (Moé) in Québécois French.
[1] https://en.wikipedia.org/wiki/Language_policy_in_France#Mino...
Furthermore, there isn't such a thing as a single "British" accent, there are a huge range of accents in Britain, just like there are a huge range of accents in the US.