Phonetic changes ("sound changes") are generally arbitrary (or done under the influence of other languages, but we'll just say arbitrary for now, at least on the surface, and definitely for our needs as well) and this is fairly uncontroversial. Phonetic changes are also unavoidable, and are a constant, rather than an exceptional alteration, and all speaker groups shall see their languages alter phonetically over time. This is also fairly uncontroversial as far as all linguists are concerned. This simply leads into the premise of the question.
First of all, there is no unique 'British' pronunciation, nor is there any one unique 'American' pronunciation. The respective 'standards' (as far as sociolects go) are the Received Pronunciation (RP) and General American (GA).
Now, sound changes are also a large scale sociolinguistic and geolinguistic phenomenon. When a sound change happens in one speech community or social stratum, it may spread to others, or may be intentionally adopted by other communities (this intentional adoption of a different idiom's phonetics and grammar is pretty much what happens with dialect levelling). Sound changes that happen in some small community may, with time, spread to others, and this is particularly noticeable within England, which has had a massive shift of rhotic accents to a non-rhotic form (coda rhotic dropping) in the last hundred or so years. This (non/rhoticity) is a fairly major phonetic feature, and is very noticeable and well-known even amongst laymen. Curiously (for many people at least; sadly it's lost charm to me), the English 'accents' from which North American English stems were, in phonetics, somewhat closer to their American counterparts than their English ones, especially with regards to rhoticity and diphthongs.
Both consonants and vowels are subject to phonetic change, and may influence one another just as easily as change on their own. The resulting noticeable differences that lead people to delineating and clearly cutting 'accents' apart are just the culmination of many small, individually inconspicuous changes. The source and extent of these changes may be obstructed by many factors as speakers interact with one another and with speakers of other groups, and each 'dialect' is the result of a multitude of interactions and shared changes. The dialects in America just happen to share more of those changes with one another than they do with the dialects of English outside of North America (for understandable, geographic reasons!)