It’s possible that you think the Puritan colonists who settled the US east coast departed from Plymouth, but that’s only partly right. It turns out their journey was already fraught by the time they got that far.
The Mayflower (together with the Speedwell) originally departed from Southampton with the Pilgrim Fathers (and presumably some Pilgrim Mothers as well) – Elizabethan religious misfits whose departure probably went unremarked at the time. After stops at Dartmouth and Plymouth to repair storm damage the Mayflower went on to the New World – the Speedwell was too damaged to make the voyage. And the rest is history. More details can be seen on the plaque at the bottom.