.
Perhaps much of the confusion, is brought about by pure semantics.
Both groups were "pilgrims", in that they both went on long, hazardous trips for religious reasons.
Perhaps I am being too technical when I insist on the correct terminology, since only the Puritans wanted to "purify" the Anglican church.
The Separatists could care less what happened to the Anglican church. To put it in modern colloquialism, so far as the Separatists were concerned, the church devised by Henry VIII could go "pound salt" !
..But of course, the Church of England, saw it differently..preferring instead to pound the Separatists.

Problem is, many current writers and talking heads use the term Puritan universally, without considering the history involved.
An interesting sidelight... the Speedwell incident.
The Separatists actually started out with 2 ships, the Mayflower and the Speedwell. Both were craft on their last legs, nearing unseaworthiness.
The Speedwell had a long life as a coastal freighter..as such, and since it wasn't destined to face the winds and waves of the open sea, as indicated, the speedwell was built for speed..quick trips between European ports.
For this reason, it was fitted with much taller masts than would be mounted on an equal sized ocean-going ship.
Out on the open sea and facing huge waves and strong winds, the tall masts worked as giant levers, tearing apart the very ribs and structure of the ship.
By this time.. in mid ocean, some pilgrims were not near so eager as previously, so when the water became calm enough, the two ships drew near each other..and exchanged some passengers.
The Speedwell then went back to England..the Mayflower continued on to the New World.
A similar situation happened to the Mayflower later on..from whence came the story of how John Alden used the screw from a printing press to secure the frame.
Since between the two ships, more chose to go on, than chose to go back..which added to the crowded conditions on the Mayflower.