| There is often an implicit assumption that daughter communities innovate and the mother community keeps the original speech. But in the 400 years since London and New York began to diverge in accent, both have undergone 400 years of evolution.
A mother community (such as the homeland, England) being large and its colonies being small (at first), you might expect different rates of propagation of change. Well, I don't know how language change arises, and I don't think any linguist has much idea, but rather than big communities being conservative because there are so many people who have to accept change for it to get established, it seems to me there may be an opposite effect: if a change is spontaneous like a genetic mutation, then the more people there are, the more mutations have a chance to arise and be taken up by a subcommunity. (The situation is similar to, but might not be the same as, this question in biology.) These days most of the accent differences seem to be in big cities.
Many of the differences between American and Southern British are because of innovations in Southern British. The sound r was lost (except before vowels) somewhere over 200 years ago in London. This change spread out and is now established all across England except the south-west and East Anglia, and is also true of the Southern Hemisphere countries colonized in the last 200 years. So American resembles Irish in being a rhotic accent (one having r everywhere) because they are both survivors of the original situation that 300 years ago prevailed everywhere in England.
However, American has innovated on exactly how it pronounces its r: instead of being a simple consonant as in Irish English, it is now a colouring of the preceding vowel: see my How to pronounce an English "R" for details.
Again, the American can't with the same vowel as can reflects the original English, and it's England that's innovated to the ah vowel. This can be approximately dated by noting that Australia etc also say kahnt; but similar changes in dance, plant happened in England after the settlement of Australia, where they have not been taken up: the can vowel in common between American and Australian dance is also what was said in London 200 years or so ago.
One mutation which is characteristic of American is the "breaking" of the vowel in can and its fronting effect on the consonant, so that it is like kyean. Now this might be a new mutation, or it might be present in some of the dialects and accents of Britain that gave rise to it; I haven't the materials by me to tell. Bear in mind also that immigration from Britain to America was continuous over many generations, and new influences and changes could keep arriving. Language change is not purely divergent like a tree: changes arising in one place often propagate across dialect boundaries. Historically, London changes gradually influence most of the rest of England, and beyond; and in the last century American has achieved a similar driving position.
This is a very quick survey, of such a level of generality that I can completely ignore the differences within American English (see Cletus the Foetus's
write-up there for details); and the possible influence of other languages such as Yiddish.
The prestige Boston accent, which is non-rhotic like London, is said to be a 19th-century innovation in imitation of London. However, I think this is an urban myth, and there is also a non-rhotic lower-class New York accent, which is highly likely to be native, not imitating England. See the first few entries in http://itre.cis.upenn.edu/~myl/languagelog/archives/2003_11.html for more detail.
Considerable detail can be found in The Cambridge History of the English Language Volume 6: English in North America, a book I haven't seen, but there's a good thorough review of it at
http://cf.linguistlist.org/cfdocs/new-website/LL-WorkingDirs/pubs/reviews/get-review.cfm?SubID=4807 | 
