I am a bit confused of the qualitative behavior of the two methods. Consider quadratic case, start by having points $x_i$, where I know the value and points $y_j$, where the values to be found. If I want to do Lagrange interpolation at any point $y_j$, I find three points from $x_i$ with values close by and fit the parabola between those three points with values $f(x_i)$. Once I have the parabola I estimate the value at the unknown point as a value of the parabola at that point. Since the polynomial is of second order I will match all the terms up to 2 in the Taylor expansion and locally will have $O(h^3)$ error.
On the other hand, for the Hermite cubic spline I fit the parabola between every two points keeping the derivatives equal at all nodes from both sides. Again, I expect the local error to be $O(h^3)$.
So, is there a difference between the two in the error? If so, why one would prefer one method to the other? Thanks!