Although UrsaCalli79's answer is very effective at explaining this mathematically, I will attempt to explain it in layman terms.
You do not leave either Newtonian theory nor Relativistic theory, it is only that one serves better to mathematically predict the forces. This can be seen in the similarity of the equations. Newtonian theory can predict the forces with a fair amount accuracy, and Relativistic theory with more accuracy.
What UrsaCalli79 mathematically demonstrated was that as you take distance from the Schwarzschild radius $-$ which also can be said to be the start of black hole's singularity $-$ the Newtonian equation can more accurately describe the forces.
To end, this essentially means there is a difference of how accurately one theory is versus the other, this difference in accuracy decreases as $r$ increases.