If you take as a concrete example the affine space $\mathbb{R^2}$, the vector subspace $V = \{(x,y)|x=y$ and $x,y \in \mathbb{R} \}$ and the affine point $(2,3)$, you get the affine subspace $\{(x+2,y)|x=y$ and $x,y \in \mathbb{R} \}$.
I used this example to imagine the sum of an affine point and a vector as an affine point and am comfortable with this. However, I am still having troubles trying to imagine the difference of two affine points as a vector. Could someone illustrate this with my example? I illustrated the sum of a vector and a point as translating the point in $\mathbb{R^2}$ using all the vectors of the vector subspace.
Note: I don't want a proof or anything, just a visualisation or some argument as to why this is logical, I know it follows out of $x \in Y = p + W \Rightarrow x = p+w => w = x-p$.
Thanks in advance!