Is computing natural gradient equivalent to deriving directional derivative? It seems to me that natural gradient is simply derived from directional derivative. For example, for a vector $v$,
$\tilde{\nabla} f \cdot v = G^{-1} \nabla f \cdot v = \lim_{h\to0} \frac{f(x+hv)-f(x)}{h}$
If they are equivalent, why do we need two terms for the same thing?
References:
Directional derivative (https://en.wikipedia.org/wiki/Directional_derivative)
Why Natural Gradient? (http://www.yaroslavvb.com/papers/amari-why.pdf)
Natural Gradient Works Efficiently in Learning (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.452.7280&rep=rep1&type=pdf)
 A: I think the difference is a bit subtle, but natural gradients are suppose to be an intrinsic properties of the function $f$ and do not depend on the parameterisation of $f$. In particular, the intuition is we want to know the curvature of $f$ independently of a choice of coordinate system. Whereas directional derivatives as defined by your definition require specification of a vector $v$ and so also are explicitly calculated with respect to vector space and a coordinate system centered around $x$. 
Natural gradients don't need a function $f$ in general to be specified, only a geometric structure (usually in this context a Riemannian manifold). To actually do any calculations we still need to choose a coordinate system. In that example of Amari's of calculating the Riemannian metric tensor $G$ when the geometric structure being studied is $\mathbb{R}^2$, to obtain an explicit form for $G$ we need to choose coordinates. Since we are studying  $\mathbb{R}^2$ with respect to polar coordinates, the Riemannian metric tensor is not constant. To put it another way, polar coordinates introduce an artificial curvature on $\mathbb{R}^2$ even though the underlying space is intrinsically flat.
People with a background in differential geometry (like Amari) seem to really care about these coordinate-free, intrinsic definitions. Sometimes it is not possible to have global coordinate system for geometric objects that they study. My impression is that the geometric structures of distributions which arise in statistics there is often a natural and important global coordinate system.
EDIT: 
It seems directional derivative on a manifold is not well-defined. You generally have two choices: Lie Derivatives or Affine Connections.
Without fully learning DG, the main thing to realize is that it is reasonable to prefer methods and algorithms that gives the same answer regardless of the representation of the problem. Mackay (2003) Ch. 4 calls these methods covariant. DG folks would say coordinate-free. I think (please somebody chime in) that Natural Gradients are really Affine Connections (see https://en.wikipedia.org/wiki/Affine_connection). And I think that this is where the word "covariant" came from in this context going back to Mackay (2003) and before: From Wikipedia

an affine connection is simply a covariant derivative or (linear)
  connection on the tangent bundle

Typical setting in ML is the manifold of distributions under some representation (the weights) and usually something like the KL divergence as a target loss.
A: From the definition of the directional derivative
$$ Df(x, v) =  \lim_{h\to0} \frac{f(x+hv)-f(x)}{h} $$
it is not obvious that $Df$ should have any special structure.  On the other hand, the mapping
$$ v \mapsto \nabla f(x) \dot v $$
has a lot of obvious structure - it is a linear map.
The point of these two definitions is that the gradient exists.  I.e., we can only have a thing called a gradient vector if the directional derivative is a linear map.  It didn't have to be, we could have found ourselves in a universe where directional derivative have much worse behaviour, and mathematics would be a lot more difficult. 
