On September 6, 1620, our Pilgrim Fathers and Mothers set sail from Holland, where many Puritans had fled, to England to furnish the boat and pick up more passengers, and headed to what was called "North Virginia" - New York harbor, specifically.
They left too late in the year. The leaky Speedwell slowed them down, and the Mayflower herself was an old tub. On November 9 they made landfall in Cape Cod (a mere 2 degrees off course), but found heading south to NY was treacherous with the autumn storms, so they gave up that effort and returned to the Cape, anchored in Provincetown Harbor, and began exploring Cape Cod (and stealing caches of Indian corn) until deciding on Plymouth as the spot to settle down for the very hard first winter.
Only 50 of the 110 on board the Mayflower survived the first winter. Had they anticipated that catastrophe, they never would have left Europe. Samoset and Squanto appeared in March (Squanto spoke English, and had already been to England, and probably to Spain too), and helped them figure out how to live, farm, hunt, and fish, in rugged New England. Plymouth, fortunately, had many large, abandoned Indian corn fields so it wasn't too difficult to get the spring planting underway.
How differently history might have developed had they ended up where they had intended in the environs of the soon-to-be wealthy Dutch mercantile colony of New Amsterdam.