It would be more accurate to say that the Separatists were a subset of the larger Puritan movement than to say that the Separatists were not Puritans.
“There is no hard line between Separatists (i.e. the Pilgrims) and Puritans.”
Except they separated from the Puritans, ie, they no longer considered themselves Puritans.
The Pilgrims founded the Plimouth Colony, the Puritans the Mass Bay and Salem Colonies.