Say $\dot{x} = f(x)$, $x\in\mathcal{M}$, $\phi: \mathcal{M} \mapsto \mathcal{M}$.
Then $\dot{\phi} = f \cdot \nabla_x \phi $.
However, one can define a lie differentiation and write
$f \cdot \nabla_x \phi = \mathcal{L}_f \phi$.
QUESTION:
What is the point of doing this? Is there any nice things we could get from writing things in lie differentiation while hard to get in standard calculus?
The wikipedia article on Lie Derivative actually nicely answers this question: "A 'naïve' attempt to define the derivative of a tensor field with respect to a vector field would be to take the components of the tensor field and take the directional derivative with respect to the vector field of each component. However, this definition is undesirable because it is not invariant under changes of coordinate system"
It is that last sentence that is of vital importance, as much of the power of the machinery of differential geometry is in its independence from choice of coordinates (hence you can choose convenient ones for the problem at hand).