In short: Newton = gradient; Einstein = curvature.

 Newton’s picture of gravity assumes flat Euclidean space and absolute time, and gives gravity as the gradient of a potential Φ. The equation of motion is $\mathbf a = -\nabla \Phi$, and the field equation is $\nabla^{2}\Phi = 4\pi G\rho$. In this basic scheme there is no notion of “spacetime curvature.” However, phenomena like tides from the Moon and Sun—spatial variations of gravity—are treated via the second derivatives of the potential, i.e., the Hessian (tidal tensor) $\partial_i\partial_j \Phi$. Thus the geometric idea that “curvature is gravity” comes only after Einstein: general relativity describes gravity by the metric and its curvature. Newton, within mechanics, viewed forces as bending orbits, organizing them with curvature radius and centripetal acceleration, while the underlying space remained flat. In contrast, GR takes free fall as geodesic motion without force, and tides correspond to curvature components $R_{0i0j}$. In the linear limit $R_{0i0j}\approx \partial_i\partial_j\Phi/c^{2}$, elevating Newton’s second derivatives into geometric language. Historically, Newton–Cartan theory later re-expressed classical gravity using connection and curvature— a reinterpretation, not Newton’s view. In short: Newton = gradient; Einstein = curvature.