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In a manifold there is a distinction between points and vectors, but in flat spacetime this seems to disappear.

For example in Minkowski spacetime you can define a coordinate 4-vector $(x_0,x_1,x_2,x_3)=(ct,x,y,z)$ and do whatever you do with usual vectors (norm, scalar product, components transformations...). The point is that if you are in a general manifold you can't to this, because the coordinates of a point are not components of a vector.

Why is this legitimate?

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    $\begingroup$ Because Minkowski space is $\mathbb{R}^4$, which is a vector space? $\endgroup$
    – ACuriousMind
    Jan 3, 2015 at 17:09

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You can treat Minkowski space-time as a vector space as long as the coordinate transformations you consider are linear. This is certainly the case for Lorentz transformations. Under more general transformations coordinates won't transform as vectors (consider the transformation that takes you to, e.g., polar coordinates). See also this question on SE.

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