Curse words evolve rather arbitrarily, though usually with the common theme of being words for taboo things (most often sex, excrement, blasphemy, and more recently, bigotry).
I'm not posting this as an answer though, because it is certainly possible for a certain word to become extremely offensive for a specific reason, and I don't know for sure about this one.
Other denizen of Greater Anglophonia are always surprised that Americans find “cock up” unacceptable in polite company. In an American context, calling Cindy a cunt in front of strangers was so disrespectful and humiliating that I thought it was only slightly less shocking than if he'd punched her in the face, and I wondered why she didn't divorce him.
So I’d suggest that different reception of the word in the two countries is down to these different predominant uses.
@oosterwal: in all dialects of English, that’s indeed the most literal meaning of ‘cunt’.
Another word with radically different connotations in British and American English is 'fag'.
In England, it is a colloquial term for a cigarette; in the USA, it is a very derogatory term for homosexual.
What accounts for the strong disapproval of anyone using the word 'cunt' in the US, when the sentiment doesn't exist to the same extent in the UK?