What exactly is variational mechanics and variational calculus ?

Variational calculus deals with finding optimal curves and such. Things like geodesics (shortest path). I don't know what you mean by variation mechanics.

Variation mechanics : Its a branch of mechanics thats based on variational calculus.

All I know is that Lagrange is to variational mechanics what Newton is to classical mechanics.

Ah, ok. Well, Langragian mechanics and General relativity both use the shortest path between two points as the path of a particle.

Which provides a more accurate description of nature Newtonian or Variational mechanics ? Or is it that they differ only in mathematical terms ?

Langragian and Newtonian mechanics differ only in mathematical terms. General relativity differs from Newtonian and Langragian mechanics greatly.

Most of Lagrangian mech applications I do are implemented via energy principles, and actually are pretty identical to principles such as the principle of virtual work or virtual power (equilibrium stuff). I think the mathematical constructs have some added value in comparison to other stuff if you're working for example with dynamics problems or need to include something with similar mathematical aspects, like optimization using numerical methods etc. The math is really elegant, though.

Lagrangian mechanics is fundamental to quantum field theory too. In fact, all of the modern high energy theories are described by Lagrangians (of more precisely, Lagrange densities). This is mainly because the Path Integral formalism of QFT weights each possible 'path' between initial and final conditions with the expenential of the action required for the path. The prefered path is then the one with the least action.

Having studied Lagrangian mechnics somewhat by now, I must say I didn't understand the mathematical derivations used, but the methods of applications were mostly very mechanical and the differential equations obtained in the end did not seem humanly solvable.

Does your "very mechanical" refer to what field of application you were studing or just that the applications were simplistic? Classical or quantum? I think they just look far worse than they actually are, the notation can do that sometimes.