This question is still hotly debated among historians. Some argue that the changes were primarily political, in the sense that the colonies gained "home rule" and paved the way to proceed with creating their own nation. These historians claim that the social structures and the economic realities of the new nation were not that different from before the Revolution, at least after the adoption of the Constitution, which they view as a very conservative document.
Others argue that the Revolution, while not a radical social uprising, unleashed forces that conservatives could not contain. This new, radical egalitarianism broke down the traditional deferential relationships and opened the door for a nation characterized by democratic politics and social, if not economic, equality. This argument was most famously put forth by historian Gordon Wood in his 1992 book The Radicalism of the American Revolution.
Perhaps the best answer to the question would be that the Revolution did not immediately lead to social change. Indeed it actually tended to benefit elites more than common people. However, Wood's interpretation, especially when we consider the effects of having cheap land on the frontier for farmers to settle, does make sense. However, it only does so if we do not consider Indians, for whom the Revolution was an unmitigated disaster; slaves, whose situation did not change; and women, who were still excluded from politics after the Revolution. For them, the changes were not as tangible as they may have been for others.