Is Schwarzschild's solution in his original paper consistent with current solutions? I was reading the Schwarzschild's original paper where he derives the Schwarzschild metric for the first time(The english translated version found in arXiv :  On the Gravitational Field of a Mass Point
according to Einstein’s Theory). 
The derivation is not very complicated and I could understand(?) most of it. However, in the final solution (Pg.6 - Eq.14) the radial coordinate $R$ is not the spatial $r$ of polar coordinate. They are related by - $$R = (r^3 + \alpha^3)^{1/3}$$
This was surprising to me because in our GR class and also in GR textbooks I used (including Wikipedia) - this was never mentioned and the radial coordinate in the metric is used as if it was spatial co-ordinate.
Why is this so? Is the original derivation wrong?
OR
Is it that this $R$ should be considered as the spatial co-ordinate instead of $r$. If so, why?
 A: Yes, the solution by Schwarzschild is the same solution as the one we call the Schwarzschild metric nowadays. The solution for the electrostatic potential of a point charge is valid if you re-express it in spherical instead of Cartesian coordinates. The same is true if you reexpress the Schwarzschild metric in isotropic coordinates, Gullstraind-Painlevé coordinates and so on and so on. 
Coordinates do not have any direct meaning in GR, and the skill of beating the physics out of the coordinate expressions is not acquired easily. To do so, you have to consider a range of thought experiments such as various observers measuring distances by various methods such as holding a rope, laser ranging, etc. In this sense, the coordinate $r$ which we usually call the Schwarzschild radius $r$ is the curvature radius of the symmetry sphere (spanned by the angles $\vartheta,\varphi$), but it will not fall out of any simple measurement as some kind of natural "radial distance". So rescaling this coordinate locally actually is not any big issue.
