The Plymouth colony is important because it was one of the first successful English settlements on the North American continent. Along with Jamestown, it helped to establish a permanent British presence on the eastern seaboard. It came close to failing, with the initial population dropping very low, but it survived, showing that New England could be successfully colonized.
The colony is also important because it was founded for religious rather than financial reasons. The Jamestown colony was a business venture, started in pursuit of profit. The Pilgrims, however, were Protestant dissenters who wished to separate from the Church of England. Persecuted in England, they came to the Americas in pursuit of religious freedom. This helped establish the idea of America as a place of freedom from oppression.
The Mayflower Compact, by which 41 Pilgrim men agreed to the establishment of a colonial government, is also important in establishing the idea of "the consent of the governed." These men freely agreed to enter into this social contract.
Possibly most important, the Plymouth colony became foundational to United States origin folklore, primarily developed in the 19th century. The idealized tale of the Pilgrims landing on Plymouth Rock, surviving against the odds, and celebrating the first Thanksgiving with the Indians (the Wampanoags) cannot be overestimated in helping to establish an American sense of pride and identity. The fact that the Pilgrims came to find religious freedom was significant to this mythology, establishing a narrative that America was the land of freedom and opportunity, settled for reasons more exalted than profit-seeking. Few United States citizens are unaware of this foundational story. Though it has been contested in recent decades as over-representing America as white and Protestant and for whitewashing settler relations with Native Americans, it is deeply engrained into the American psyche.