Suppose there are two connections over the tangent bundle of a smooth manifold with the same geodesics. What is the relation between the curvature tensors, the Ricci curvature and the scalar curvature of these two connections? Note that the question gives no address to the metric, such as Riemannian or the Levi-Civita.
You are looking for the theory of projective connections. See the paper of Molzon and Mortensen, The Schwarzian derivative for maps between manifolds with complex projective connections, Trans. Amer. Math. Soc., Volume 348, Number 8, August 1996 which is surely the best survey. It treats the complex case, but the details are the same as for the real case. You might also look at the paper of Kobayashi and Nagano, On projective connections.