A connection Γ on Y → X is called the affine connection if it as a section Γ : Y → J1Y of the jet bundle J1Y → Y of Y is an affine bundle morphism over X.
In particular, this is an affine connection on the tangent bundle TX of a smooth manifold X.
These are related but distinct concepts both unfortunately making use of the adjective "affine".)
It is readily observed that R = R + T, where is the torsion of Γ with respect to the basic soldering form σ.
In particular, consider the tangent bundle TX of a manifold X coordinated by (xμ, ẋμ).