the upshot seems to be you should not worry about it, there is indeed no big difference.
In my experience what matt grimes says is common, i.e. a vector field is often a vector valued function whose values are taken in the tangent space to the domain, hence it is drawn as a family of tangent vectors to points of the domain.
On the other side, a vector valued function may have any kind of vectors as values. But as Arildno points out the more general concept is also called vector field by some physicists.
so you have to read the context carefully in given case.
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