If you look at the equations, relativity looks like a refinement to Newtonian physics: at small masses and low velocities, relativity makes the same predictions

If you look at the equations, relativity looks like a refinement to Newtonian physics: at small masses and low velocities, relativity makes the same predictions as Newtonian physics. However, it is not correct to call this a "small improvement." Relativity was a radical change to our understanding of how the universe behaves. In a Newtonian universe, information (including gravitation, electrical attraction, and light) travelled instantaneously. This does not happen in relativity, and nor does it happen in the universe we live in. Einstein did not merely improve upon Newton's work; Einstein proposed a theory which explained the ways in which we do not live in a Newtonian universe. Thomas Kuhn goes into this concept in detail in "The Structure of Scientific Revolutions."