Why is difference of points not a valid definition for a vector in curved space? In page-49 of MTW (1973 edtn), the following picture is shown:

After seeing this picture, the question which arose in my head is why exactly can we not define a vector as difference of points in curved space?
 A: The short answer is that vectors must satisfy the eight axioms of a vector space in order to qualify as vectors.
Flat space is professionally known as affine space, which I have described in my answer here. In affine space, one can subtract two points to obtain a vector. By defining an origin and position vectors, one can also make points behave like vectors, therefore making it a vector space.
However, on a manifold, the situation is quite different. Position vectors can no longer be defined because there is no way to make points add like vectors. Even if we smoothly label the points using coordinates $x^1, \ldots ,x^n$, where $n$ is the dimension, there is simply no way to choose coordinates such that points obey the vector space axioms. This is why the "difference of two points" is no longer a meaningful definition of a vector. Conversely, if the points in our manifold do end up behaving like vectors, our manifold is an affine space.
However, the partial derivatives along all smooth curves passing through a point on the manifold do form a vector space, which is the tangent space.
I believe this answer on Mathematics.SE (which I personally upvoted) gives a really good summary of the situation.
A: For one thing, vectors obey linear structure, i.e. we can write $\vec{v}$ as sum of two vectors $\vec{v_1}$ and $\vec{v_2}.$
So if we want to have operation $A=B+\vec{v},$ we should also demand that operations  $$A=B+\vec{v}_1+\vec{v}_2=(B+\vec{v}_1)+\vec{v}_2=C+\vec{v}_2$$ and
$$A=B+\vec{v}_2+\vec{v}_1=(B+\vec{v}_2)+\vec{v}_1=D+\vec{v}_1$$
produce the same point.
In curved space however, going first in the direction of $\vec{v}_1$ and then $\vec{v}_2$ does not produce the same point as going first in the direction of $\vec{v}_2$ and then $\vec{v}_1$. The difference between these two paths actually defines Riemann tensor, which characterizes curvature of space and is zero only if space is flat.
A: Because in curved spacetime you can't take the difference of a vector at point P and point P' unlike in Minkowski spacetime.
Imagine abovementioned two points separated by $dx^{\mu}$ and let's call the vector at point P $a_{\mu}(P)$ and the other $a_{\mu}(P')$. You can infinitesimally transport the vector at P by $dx^{\mu}$. Thus $A_{\mu} = a_{\mu} + \delta a_{\mu}$ which is a vector at point P'. Now you can take the difference, hence it becomes
$$a_{\mu}(P') - A_{\mu}(P') = (a_{\mu, \nu} - \Gamma^{\lambda}_{\mu \nu} a_{\lambda})dx^{\nu}$$
