This is a very broad question, but some general influences are easy to perceive.
Great Britain is an island nation with an established population that has not been successfully invaded for nearly a thousand years. This setting gives its inhabitants a feeling of security and uniqueness, but also encapsulation. British security and sea power gave Britain the idea that it had the right to rule large parts of the world, but there was never any impulse to make these part of Britain itself. Because it is an island, Britain has difficulty feeling itself part of Europe, and tends towards insularity. One consequence is that British culture has by and large been relatively uninfluenced by Continental trends; British people are very poor at learning foreign languages and customs and reluctant to assimilate immigrants. They have a sense of divine appointment and innate superiority, like most major cultures, but it is not expansive in the sense of expanding the territory of Britain itself, which is seen as a bounded and limited "fortress" guarded by the sea:
This other Eden, demi-paradise,
This fortress built by Nature for herself
Against infection and the hand of war,
This happy breed of men, this little world,
This precious stone set in the silver sea,
Which serves it in the office of a wall,
Or as a moat defensive to a house,
Against the envy of less happier lands
(Richard II, Act II, Scene 1)
The United States, on the other hand, is the home of the Great Frontier. It is an immigrant nation whose vastness and relatively undeveloped state appears to offer an infinity of advantages to anyone who will come to exploit them. The much greater environmental scope of the United States, when compared with Britain, gave rise to the concept of "Manifest Destiny," the physical expansion of the American state to cover the entire North American continent, and its influence to encompass the world. The initial emptiness of the land, and the need to populate it, led to the feeling that America was a work in progress, a "melting pot" which could accept many (though by no means all) different ingredients and make them "American."
Thus, different environmental conditions led to British culture thinking of itself as a finished product, a "precious stone set in the silver sea," while American culture sees itself as an enterprise that is constantly expanding and developing, shaping a national character rather than guarding over one inherited from the past.