A fireplace is a beautiful, cost effective way to warm your home, but whether you get a tri-fold or a flat fireplace screen depends on your personal preference. They both serve the same purpose: to keep sparks, ashes and other fireplace debris from spilling out onto your floor, and to take the edge off the heat coming from the flames.
The 2 primary differences between tri-fold and flat screens are convenience and decorative quality. Tri-fold screens tend to have a more artistic slant, while flat screens are more straight-forward and utilitarian. Flat screens come in decorative designs, to be sure, however they don't have quite as large a selection as tri-fold screens.
Tri-fold screens are more convenient in that they allow easier access to a burning fireplace due to the folding sides. In order to get to the fire with a flat screen, you have to move the entire thing, which can be a tricky procedure if it's excessively hot. Tri-fold screens, due to the way they wrap around the fireplace, also provide better protection from heat on each side.
Price is comparable, for the most part, however the more decorative you go, the more expensive the screen.