By 'Newton's Law of Gravity', I am referring to
The magnitude of the force of gravity is proportional to the product of the mass of the two objects and inversely proportional to their distance squared.
Does this law of attraction still hold under General Relativity's Tensor Equations?
I don't really know enough about mathematics to be able to solve any of Einstein's field equations, but does Newton's basic law of the magnitude of attraction still hold?
If they are only approximations, what causes them to differ?
Answer
Yes, in the appropriate limit. Roughly, the study of geodesic motion in the Schwarzschild solution (which is radially symmetric) reduces to Newtonian gravity at sufficiently large distances and slow speeds. To see how this works exactly, one must look more specifically at the equations.
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