Upto my knowledge there are three "kinds" of tamil speaking people:
1. Tamils from north and east, who settled around 3rd to 4th century AD
2. Muslim Tamils mostly from east and also from colombo and Ratna pura
3. Tamils from south india who were taken to Srilanka during the 18th century by British to work in the Tea Estates. (Minister Thondamaan is aone of them)