Turnscrew is the name from long ago across the pond in England. Screwdriver is also correct...depends on your preference and how much you would like to mess with the head of someone who has no idea about tools

They are cabinet maker's screwdrivers as well as anyone else who needed one, just more often used in wood working. The flat blade and flat sides on the handle was the way that they were made through the late 1800's. Why, I do not know and have not found any firm information on this practice. Most that I have seen are made from cast steel. Stanley Rule and Level Company offered this style up through the mid 1910's as they were still popular. Around this time the round shank as we now know was starting to become more popular. They are still used, particularly in the smaller sizes, by gunsmiths since it is easier to hollow grind them for the particular screw size you are using. They can still be found today, although they are not quite in the abundance that they were years ago. Blade lengths usually ranged from 3 to 10 inches. Some of the Stanley's I own have a patent date of June 6, 1871. I also have a long Bates that has a 7-1/2" blade and an overall length of 14-1/2". The true difference : screwdriver cost = $5.00, turnscrew cost = $50.00 (a joke I once read on a gunsmithing web page)
Just my $.25 (formerly $.02, but inflation got it)