The impact of the Pilgrims on America is likely not as substantial as most America History textbooks make it. The pilgrims were not the first settlers from Britain in North America. Virginia was granted a royal charter in 1606 to found a colony, though English settlers had resided in Virginia since about 1590. The initial settlement of Roanoke was abandoned by 1590, but the Jamestown settlement was ultimately a success. Jamestown faced several rough years, due largely to placing an inordinate amount of resources into searching for exportable commodities and placing too little effort on growing food. After learning to grow tobacco (as well as the necessary food) the settlement began to flourish. Tobacco became the cash crop of Virginia and, in order to maximize profits for the crown, led to the importation of African slaves because Native American slaves were unsuitable for the task. The first African slaves arrived before the Pilgrims landed on Plymouth rock.
What would later become the New York colony was initially settled by the Dutch before the Pilgrims landed on Plymouth rock as well. The Pilgrims, therefore, were not the initiators of colonization nor did they drive colonization north of Virginia. However, the Pilgrims did have two impacts on how the New England colonies developed. First, the Puritans demonstrated a religious intolerance that is surprising considering the reason for settling in North America was religious intolerance against Puritans in England. This religious intolerance ultimately led to the founding of Delaware by dissenters. The religious intolerance was also connected to the importance of religion in the legal and governing aspects of the Massachusetts Bay colony. This helped prolong the impacts of the Salem Witch trials (as well as nearby witch hunts), though it was not the leading cause.
The second impact was the governing style of New England. Because the Pilgrims landed outside of land that belonged to the English Crown and established a self-governing settlement under the Mayflower Compact prior to obtaining a royal charter, settlers in the Massachusetts Bay Colony retained a substantial amount of involvement in the political processes compared to other colonies.
These issues likely had little long term or significant impact on the development of America, though they do provide useful historical studies.