Pilgrim can identify anyone looking for a Holy Place or seeking religious liberty such as the Pilgrims that came across on the Mayflower to Massachusetts as Separtists establishing Congregationalist churches in the new land.
John Smith was an explorer that settled Jamestown in Virginia. Religion wasn’t really the focus of his exploration. The only thing in common he had with the Pilgrims was the original naming of Jamestown as New Plymouth because his departure port was Plymouth, Devon which was the same port the Pilgrims departed from. The Pilgrims named the their landing location the Plymouth Colony which differed from John Smith’s exploration of Virginia.