Exactly! However Newton's equation doesn't include densities - but GR does. And it makes me wonder, why science continues to use the Newton's equation, if it contradicts in some way the GR field equations.
A very valid question. You see, even though GR is the more comprehensive and accurate model of gravity, many of its relativistic effects and corrections become relevant only in situations where we deal with strong gravitational fields, high velocities, or forms of energy other than mass. On the other hand, in cases where only weak gravitation and no other forms of energy are involved, relativistic effects are so small as to be negligible for most purposes.

If you look at it in terms of computational effort, you will see that Newtonian gravity is a very simple model, and many problems can be solved by the pen-and-paper method, i.e. algebraic manipulations done manually on a sheet of paper, or via standard CAM software. This is why it is still taught in high school physics. GR on the other hand is a very different story - it is the more precise and accurate model, but in all but the most simplistic of cases the Einstein equations are extremely difficult to solve; even numerical methods (which do exist) require substantial computing power.

You can probably now see the answer to your question - Newtonian gravity continues to be used in cases where relativistic effects can be neglected, simply because it is much easier to handle and solve, mathematically speaking. For example, if you look at the Earth - which has weak gravity in the bigger scheme of things -, you will see that Newtonian gravity gives very accurate predictions under most circumstances. On the other hand, if you look at the gravity of our sun (or even stronger sources), you will notice that Newton gives predictions that are ever so slightly off, so GR is needed. You get the idea.

Formally, Newtonian gravity is the asymptotic limit of GR - for example, in the Schwarzschild metric, if you make

*r* very large, you will recover Newtonian gravity. Hence, GR and Newtonian gravity aren’t really different and separate - Newton is a subset of GR, that is still valid under certain restricted circumstances, and hence continues to be used due to its simplicity and ease of application.