Difference between path and vector field 
What is the difference between a path and a vector field?  

From what I understand the unit vectors $\mathbf i$, $\mathbf j$, and $\mathbf k$ are actually vector fields (constant vector fields to be exact).  Then if we have a path $$\mathbf r(t) = x(t)\mathbf i + y(t)\mathbf j + z(t)\mathbf k$$ it's just a linear combination of vector fields.  
$1)$ So doesn't that make it a vector field as well?  
$2)$ So then are paths just a specific type of vector field or are they different concepts?
 A: A vector field assigns a vector to every point in the space, a path only assigns a vector to a subset of points consisting of a curve of some sort.
A: 
the unit vectors $\mathbf i$, $\mathbf j$, and $\mathbf k$ are actually vector fields  

Not really. These vectors are elements of $\mathbb{R}^3$. An element of $\mathbb{R}^3$ is not the same as a map into $\mathbb{R}^3$. 
Now, one can consider a constant map into $\mathbb{R}^3$ which takes the value $\mathbf i$. But... there are many such constant maps, one for each set imaginable.  There is 


*

*the constant map from $\mathbb{Z}$ to $\mathbb{R}^3$, which takes value $\mathbf i$ on every integer

*the constant map from $\mathbb{R}$ to $\mathbb{R}^3$, which takes value $\mathbf i$ on every real number

*the constant map from $\mathbb{R}^2$ to $\mathbb{R}^3$, from $\mathbb{R}^5$ to $\mathbb{R}^3$... from $[0,3]^3\times \mathbb{Z}\times \mathbb R$... 


So, which one of these "is actually" $\mathbf i$? Probably none. 
In some contexts it may convenient to use the same letter $\mathbf i$ for the constant map $\mathbb{R}^3\to \mathbb{R}^3$ that takes the value $\mathbf i$ everywhere. But it should be understood that this is using the same letter for a different, though related object. (A convenient abuse of notation.) 

So: no, these are different things. A path is a map from $\mathbb{R}$ to $\mathbb{R}^3$. 
A vector field is a map from $\mathbb{R}^3$ to $\mathbb{R}^3$ (actually, this is also an abuse of notation, which will become obvious later, when you   study vector fields on manifolds). 
These are very different concepts, and not just because the domains are different but because of what we do with them. 
