What is the "flat back" you refer to?
"Flatback" and "stocked to the fences" are two major variations of the joint between the action and the butt stock. "Flat back" has a short horizontal extension of the action back and joins the stock in a more or less straight vertical line. "Stocked to the fences" does away with the short action back extension and brings the stock right up to the fence/ball.
FB was used a great deal during the 1880's - 90's, but got demoted generally to lower Original Quality grade guns as the "London bestr gun" pattern gained fashion status. It is common to find "best work guns" in FBSLE form the late 1800's, less so from the early 1900's.
As far as I can determine, there is no overt design advantage to either. There is undeniable fashion status advantage.
A Charles Hellis stocked to the fences SLE.