So basically I've been studying some basic differential geometry and I've learnt that a smooth map of the form $\gamma : \mathbb{R} \longrightarrow \mathbb{R}^{3}$ can be interpreted as a map that takes the real line and smoothly embeds it in $\mathbb{R}^{3}$, with the result being some smooth curve in $\mathbb{R}^{3}$.
Similarly a smooth function $\phi : \mathbb{R}^{2} \longrightarrow \mathbb{R}^{3}$ can be interpreted as taking the 2 dimensional Euclidean plane and smoothly embedding it in $\mathbb{R}^{3}$ with the result being a smooth surface.
Now I'm assuming that a function $F: \mathbb{R}^{3} \longrightarrow \mathbb{R}^{3}$ can be interpreted as a smooth distortion of space but a smooth map of this sort is what we usually interpret as as a vector field on $\mathbb{R}^{3}$ (assuming $\mathbb{R}^{3}$ is equipped with a vector space structure).
To pick a slightly physics oriented example, can a vector field like the Electric Field be thought of as a smooth distortion of space? If so then how would we interpret vector field operations like the divergence or curl?
All of this seems hugely unintuitive to me and I'm not even sure that what I said makes any sense so a little guidance would be hugely appreciated.
The reason you are confused is that Euclidean space carries a natural "connection," which makes it possible for us to view points as vectors and vice versa. In a general manifold this ambiguity is not possible.
As Moishe Cohen points out, the two pictures are very different. One way to view a smooth map from 3-space into itself is as a smooth volume distortion. (If the map is a diffeomorphism, you in fact just get a smooth change of coordinates.) Another way is to view such a map as assigning a tangent vector to each point of space. This point of view is only possible becaus the standard connection gives a canonical isomorphism of each tangent space $T_p\mathbb{R}^3$ with $\mathbb{R}^3$.
The distinction is clearer in the setting of general manifolds. Let $M$ be a smooth manifold. A smooth vector field on $M$ is a smooth assignment of a tangent vector to each point of $M$. That is, a smooth vector field smoothly assigns to each point $p$ a tangent vector in the tangent space $T_pM$. More formally, a vector field is a smooth section of the tangent bundle, i.e. a smooth map $v:M\to TM$ such that $\pi\circ v=\text{Id}_M$. Note, in particular, that a vector field is NOT a map from $M$ into itself. When you think of a map from 3-space into 3-space as a vector field, you are really using the canonical affine connection on $M=\mathbb{R}^3$ to identify each tangent space $T_pM$ with $M$ itself.
The upshot is that a vector field is a map $M\to TM$, whereas a smooth distortion of points is a map $M\to M$. (The two types of maps aren't totally unrelated. As Alfred Yerger hints at, a vector field on a manifold induces a flow, which is a one-parameter family of diffeomorphisms $M\to M$ given by flowing along the integral curves of the vector field.)