In both "The Legend of Sleepy Hollow" and "Rip Van Winkle," part of what makes America so unique is the degree to which it is depicted as a country in transition, with its colonial roots giving way to post-Revolutionary War republicanism. In this sense, in Irving's depiction, it is simultaneously very, very old (still being connected to its European roots), yet also vitally new, perhaps even unprecedented from a European perspective.
In Sleepy Hollow, Irving presents a setting that is itself suspended between these two epochs of American history, belonging to both of them at once. This story is set after the American Revolution (most notably, there is the example of the Headless Horseman, a Hessian soldier who died during the war). Yet Sleepy Hollow's prevailing culture and way of life has remained largely unchanged since the earliest days of Dutch colonization (from whom its then current residents descend). In this sense, the colonial and post-colonial are portrayed as coexisting with one another, straddling both sides of the Revolutionary divide.
"Rip Van Winkle" depicts this same transition in a different way, as it tells the story of a man who sleeps for a span of twenty years, through the entire history of the Revolutionary War. Here it is Rip himself that is suspended between these two worlds, returning to a village he finds unrecognizable, one that has been (since he fell asleep) animated by the intense spirit of civic republicanism awakened by the Revolution.
In both stories, this sense of transition is a key part of what makes the United States' of Irving's own time so unique. These stories are, in a way, about the United States emerging out of its colonial past to find a new identity for itself.