Let X be a vector field on a smooth manifold M and $ f \in C^{\infty}\left(M\right)$.
I want to know about the difference and the intuition one should have about the following two maps:
The map $Xf \in C^{\infty}\left(M\right)$ defined by $Xf(p)=X(p)(f)$ (by definition, tangent vectors are certain maps $v:C^{\infty}\left(M\right)\to \mathbb{R}$)
The map ${L}_{X}f \in C^{\infty}\left(M\right)$ defined by ${L}_{X}f(p)=df(X(p))$ (where we canonically identified $T \mathbb{R}$with $\mathbb{R}$)
The underlying definitions and notations I used come from the book "Introduction to Smooth Manifolds" by John M. Lee, see here https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth_manifolds.pdf.
This is what I have been thinking so far:
$Xf(p)$ is the value of change of $f$ in the direction of the vector $X(p)$, i.e. the map gives for each point $p \in M$ the directional derivative of $f$ in direction $X(p)$.
I am less sure about what to think of the first map inuitively. I would guess the same thing as the first map, but without being exactly sure why.
You can see that $\mathbb{R}$ is a manifold with a global coordinate system, the global chart is the identity function.
if $f$ is a real valued function on $M$ let $p \in M$
then $df_p : T_pM \mapsto T_{f(p)} \mathbb{R}$
so $df(X)(p)=df_p(X(p)) \in T_{f(p)} \mathbb{R}$
if you compute $df_p(X(p))$ in coordinate you have $df_p(X(p))=df_p(X(p))(id_{\mathbb{R}}) \frac{\partial}{\partial_x}|_p$
Since you have a global coordinate system you can identify $T_{f(p)}\mathbb{R}$ with $\mathbb{R}$ and $df_p(X(p))$ is represented by its coefficient $(df_p(X(p))(id_{\mathbb{R}}) =X(p)(id_{\mathbb{R}} \circ f)= X(p)(f)$
hence you have $L_Xf(p)=Xf(p)$