British victory was not as important as American participation in that victory in foreshadowing future events. Americans had a new sense of patriotism, but also a sense of Americanism. They had a new vision of their importance to the Empire, so much so that Benjamin Franklin spoke of a time when the capital of the empire would be on the Hudson rather than the Thames. There was growing resentment and distaste by Americans for the British military in response to the haughtiness of the British towards the Americans, most of whom were volunteers, and the complete ineptitude of the British at frontier methods of fighting. Recent historical research has indicated that American fighters in the war felt a moral superiority to their British counterparts. The harsh punishments imposed on British soldiers and the cursing, drinking and whore-mongering of the redcoats contributed itself to this feeling of superiority. This situation was exacerbated when George Grenville, Chancellor of the Exchequer, was determined to keep British troops in America even though they were neither wanted nor needed there. Grenville did not want influential British officers meddling in British politics, so he kept them in the Americas, contributing to the growing feeling of resentment.
The immense cost of the war was also a factor, as well as the issue of governance of Western lands. The new monarch, George III imposed the Royal Proclamation of 1763 on the colonies, forbidding settlement west of the crest of the Appalachian mountains. This proclamation was not only largely ignored; it contributed to a growing sense of independence and "separateness" from the remainder of the Empire. In short, there was a growing sense that the colonists were Americans, not British. All of these were a result of the French and Indian War.
I would simply add that the removal of the threat of the French in Canada allowed the conflicts mentioned in the answer above to boil to the surface by reducing the necessity of the protection of the British Empire that the French threat had required. The British also took more interest in exerting direct political and economic control of the colonies.