Coreset and VC dimension
Samples provides you only with statistical guarantees. For a fixed $\epsilon$ whatever you compute would hold for everything "except" $(1-\epsilon)$ fraction (I am being very informal here). Thus, taking an $\epsilon$-net sample of a set of points in the plane, computing its smallest enclosing disk, results in a disk $D$ that contains $(1-\epsilon)$ fraction of the whole point set (or $(1-\epsilon)$ of the measure).

Coresets on the other hand, provide geometric guarantees. For example, for a point set $P$ in the plane, an $\epsilon$-coreset for smallest enclosing disk is a subset $S$, such that any disk $D$ that contains $S$, if you slightly expand it (formally, scale it up by a factor of $(1+\epsilon)$, the it must contain all the points of $P$.

The beauty of samples is that you can generate them without a priori knowing what you are going to use them for. Coresets on the other hand are application dependent - you need different algorithms to compute them depending on the function you are trying to capture.

As for characterization - we have many positive results (see the survey here: http://sarielhp.org/p/04/survey/). We also have counter examples which are somewhat mysterious - http://sarielhp.org/p/02/2slab/ (no coreset no cry [after working for months on trying to compute a coreset for this problem, I definitely wanted to cry]). The rule of thumb is that you are not going to have a coreset if:

(A) The underlying function change dramatically if you can remove a single point, and there are many such points (say, closest pair, or k-center clustering, etc). Solving such problems in linear time requires different techniques.

(B) Or putting (A) differently - the underlying function is not very stable if you move points around.

It is unlikely to have a better characterization without assuming something significantly stronger on the underlying function that one is trying to compute.