Sawhorse projections generally show when something is antiperiplanar or synperiplanar, more easily than something like a Newman projection or a basic line structure can.
Take ethane as an example.
An antiperiplanar conformation has a #180^@# dihedral angle, i.e. the atoms of interest across one bond are on opposite sides along the vertical molecular plane.
A synperiplanar conformation has a #0^@# dihedral angle, i.e. the atoms of interest across one bond are on the same side along the vertical molecular plane.
A sawhorse projection approximates this 3D structure extremely well, and allows one to judge whether an #E2# reaction is likely to occur or not (it requires an antiperiplanar conformation).
A Newman projection would depict the dihedral angle correctly, but because one would be viewing the important atoms from the front instead of an aerial view, you might actually be looking at an octahedral molecule (with a main-chain bond length of #0#), instead of, say, a two-carbon organic molecule.
Furthermore, a #180^@# dihedral angle does not really tell you what the bond angles are of the #"X"-"C"-"C"-"Y"# bond, where #"X"# is the atom at the top middle of a Newman projection and #"Y"# is the atom at the bottom middle of the same Newman projection.