having read the OPs questions a bit more, he seems to be asking very basic questions. In answer I would say that there are three increasingly structured spaces involved, 1) affine space, which is simply the physical space in which bodies have place. It makes sense here to draw straight lines and determine parallelism, but there are no coordinates and no algebraic operations.
2) a vector space which looks exactly like affine space except for the extra structure of an origin, a fixed point. Now one can make algebraic operations like adding vectors using the origin and the parallelogram law. I.e. now each point represents a vector by drawing an arrow from the origin to that point. And given two arrows starting from the origin, we can complete that angle to a parallelogram and then the diagonal starting from the origin is the sum of the first two vectors. There are still no coordinates.
These two spaces are related, since we may use the points of the original affine space as the points of our vector space, just with the extra data of having chosen one of them. This is often done, and then one acts as if the points now have either of two possible identities, points or vectors. I.e. even after being given an origin O, if we look at another point P. we can use that point to represent the location of an object, in which case P is being thought of only as a point of the original affine space, or we can think of P as representing the displacement vector OP, representing perhaps the constant velocity required to go from O tp P in unit time.
Now in this setting some confusion still can arise, e.g. if we want to represent displacement vector PQ between two points other than the origin, as a vector. It is natural to use the arrow PQ from P to Q, but this has not been defined as a vector in our space, since it does not begin at the origin. So technically, we must represent PQ by the arrow OR starting from the origin and parallel and similarly oriented to PQ. Then PQ and OR will be opposite sides of a parallelogram. Another approach is common if a little abstract, and that is to just agree that all arrows in our vector space represent vectors, but the two arrows are considered equal, or equivalent, if they are parallel and of same length. Now things are a little confusing, namely what is a vector in our vector space, a single point P, representing the vector OP, or is it an arrow PQ? This confusion stems from the dual nature of our space, namely it was originally an affine space of points, but now is also thought of, with its origin, as representing vectors.
This situation, although confusing, actually has some advantages, since we can make algebraic computations in the space using vector algebra, and this is useful. The confusion is what to consider a given object, a point or a vector? For this it helps to know the physical object beimng represented, i.e. is it a position, or a velocity/ displacement? For this reason, some people like to define the vector space associated to the affine space as determined by ordered pairs of points, to keep them separate from the actual points of the affine space, or to consider them even more abstractly, but naturally, as translations of that affine space. The problem still arises to calculate with those translations, and for that representing them by pairs of points or arrows is helpful.
3) Coordinate space. This is affine space together with an origin and also a set of axes with units marked on them, i.e. it is a vector space together with an ordered basis of that space. Now one can assign numbers to points and to vectors and can actually carry out calculations. Because this is so useful, it is common to use this coordinate space to reprsent both the affine space of points and the vector space of velocities. Now the user has to struggle to remember in each situation whether he is dealing with points or vector concepts, and remember as well that the actual numbers associated to them are somewhat artificial as well, depnding on choices of units. I.e. if two points with coordinates are both representing position vectors, it probably makes no sense to add their coordinates.
so there are three spaces, the affine space where bodies live, the associated vector space of translations on that space, and the more refined numerical coordinate grid which we carry around and impose on either of these to make calculations. This is all Newtonian physics, i.e. based on (coordinatized) Euclidean geometry. If we want to move into a consideration of curved space, we need to introduce a new space, maybe curved like a sphere, and now at each point we have a different vector space approximating it, where the velocity vectors acting on that point live. In a sense we just "warp" the original affine space, losing its affine qualities, but we keep the vector spaces, but consider there to be a separate vector space at each point. I.e. now it is not possible to equate PQ with any OR, since there is no translation from P to O. In this setting the distinction Fresh emphasized is that the points live in the curved space, and the vectors live in the vector spaces attached tangentially at each point of the curved space. Of course one can use differential geometry to introduce "parallel translates" and make further metric assumptions and conclusions, but I understood your question to be about pre-relativistic phenomena.
Anyway it is a wonderful question. Apologies for the long and possibly unhelpful answer. Actually I have tried now to answer all 3 of your original questions in some form. In particular, 1: what is an affine space, 3: why are vector and affine spaces different, and 2: is a vector space already a coordinate space (no).