Difference of affine connections is a tensor ($\Delta (X,Y)={\nabla }_{X}Y-{\tilde{\nabla }}_{X}Y$)

The difference of two affine connections is a smooth tensor field of type $(1,2)$. Linearity over smooth functions holds in the first slot by the connection axioms. In the second slot, the two extra terms $X(f)Y$ cancel, so $\Delta (X,fY)=f\Delta (X,Y)$. This tensoriality makes the value depend only on $X_p,Y_p$. In coordinates its components are the differences of the Christoffel symbols, although either collection of connection coefficients individually is not a tensor.

Two affine connections have exactly the same geodesics with the same parameters if and only if their difference tensor vanishes on every diagonal pair $(v,v)$. Indeed, their covariant accelerations differ by $\Delta (\dot{\gamma },\dot{\gamma })$; starting a geodesic at each arbitrary initial vector proves necessity. Polarization then says that the symmetric part of the difference is zero. This condition is stronger than agreement of unparametrized geodesic equations.

The difference of two torsion-free connections is symmetric, because subtracting their zero-torsion identities gives $\Delta (X,Y)=\Delta (Y,X)$. Agreement of parametrized geodesics makes this difference skew-symmetric as well, by parametrized geodesics determine the symmetric part of an affine connection. Over the real numbers it must therefore vanish. Without the torsion-free condition, an arbitrary skew-symmetric difference changes torsion without changing the parametrized geodesics.