What does Differential Geometry lack in order to "become Relativity" - References

In (special) relativity $\mathbb{R}^n$ comes with a different metric, with signature $(-,+, \ldots +)$.

The mathematical study of manifolds with a Lorentzian metric is called lorentzian geometry, or pseudo-Riemannian geometry or semi-Riemannian geometry (the latter are more general as it refers to any metric which is not definite positive).

A generic Lorentzian manifold is still not a spacetime model in General Relativity, for that you also need it to satisfy Einstein equations $G=8\pi T$.

Mathematical references are O'Neill "Semi-Riemannian Geometry With Applications to Relativity" Beem, Ehrlich "Global Lorentzian geometry"

Initially ignore the effects of spacetime curvature (general relativity). Consider a homogeneous (flat) real four-dimensional space. If the distance between a pair of points is given by a definite (Euclidean) quadratic form, the symmetries of space include the usual orthogonal symmetry group $\mathrm{O}(4)$ and translations (producing the Euclidean group $\mathrm{E}(4)$). The only difference in going to special relativity is that the quadratic form is indefinite (Lorentzian), which results in the indefinite orthogonal symmetry group $\mathrm{O}(1,3)$ and translations (producing the Poincaré group). Rotations just behave a little differently, but should still be thought of as a symmetry group of four dimensional space.

Everything about parameterization of curves and velocities follows directly from this - indeed, all of the geometry of special relativity follows. A pitfall to avoid is thinking of time as universal; it is specific to the coordinate system. If you extend this local picture to differential manifolds, and you add Einstein's equation that relates the curvature of spacetime to its content (specifically the stress-energy tensor), you have general relativity.

As an aside, just as dropping the parallel postulate of Euclidean geometry produces elliptic and hyperbolic geometries, Einstein's postulates for special relativity allow for "non-flat" (de Sitter and anti-de Sitter) geometries, with symmetry groups $\mathrm{O}(1,4)$ and $\mathrm{O}(2,3)$ respectively.