One of the most apparent differences between North American English and many (but not all) British dialects is the pronunciation of syllabic final 'R'. Most American accents are rhotic (except in some areas in the Northeast and eastern seaboard like New York and New England and in the South like parts of Georgia and New Orleans) and speakers pronounce the 'R' in words like, "bar", "car", "bird" etc. Many English accents (but not necessarily all of them), as well as the accents of Australia and New Zealand tend to be non-rhotic, meaning the 'R' at the ends of syllables are silent.
https://en.wikipedia.org/wiki/Rhoticity_in_English
How did this situtation arise? Here's a tantalising clue, Elizabethan English in London and the South of England; meaning the language spoken during the time of Shakespeare and in which he wrote his famous plays and sonnets, was rhotic.
Shakespeare's original pronunciation sounded a little bit like how modern West Country people speak in the UK and it does sound eerily American as well. In the clip above, a comparison of modern RP and Elizabethan English pronunciation starts around 2:58 mins.
Apparently sometime around the late 18th and early 19th centuries, in London and other cities in the South of England, members of the aristocracy dropped their final 'R's and this then became the origin of a "posh" British accent. The R-dropping had already begun in informal speech much earlier but it was only around Jane Austen's time that it became a part of "proper" English.
American and Canadian (as well as many rural UK dialects) actually preserve some pronunciations that had become unfashionable among upper-class English people in Southern England more than two centuries ago.