Controlling quality of discretized region meshes

In principal you should be able to do

r = RegionDifference[Ball[{0, 0, 0}], Ball[{0, 0, 1}]];
rp = RegionPlot3D[r, PlotPoints -> 50];
DiscretizeGraphics[rp]

Unfortunately, this does not work and is hopefully improved in a future version.

One thing you can do, however, is use the finite element mesh generator for this:

Needs["NDSolve`FEM`"]
m = ToElementMesh[r, 
  "BoundaryMeshGenerator" -> {"RegionPlot", "SamplePoints" -> 50}, 
  "MeshOrder" -> 1]

You can then convert the ElementMesh to a MeshRegion simply by using

MeshRegion[m]

Note that in this case MeshRegion does not show it's output since it's beyond a threshold (too big). It has some 300000 tetrahedron elements.

You can force it to display with

Show[MeshRegion[m]]

The "MeshOrder" is set to one (= linear elements) and the "SamplePoints" correspond to the PlotPoints in RegionPlot. There is more information about boundary mesh generation in ToBoundaryMesh, ToElementMesh, ElementMesh and the related mesh generation and element mesh visualization tutorial.

One last thing I should add, is that the 3D stuff does have some rough edges.

Update version 12.1:

In version 12.1 you can use the OpenCascadeLink to do the following:

Needs["OpenCascadeLink`"]
Needs["NDSolve`FEM`"]
r = RegionDifference[Ball[{0, 0, 0}], Ball[{0, 0, 1}]];
ocr = OpenCascadeShapeBooleanRegion[r]
(* OpenCascadeShapeExpression[3] *)

bmeshOC = OpenCascadeShapeSurfaceMeshToBoundaryMesh[ocr];
bmeshOC["Wireframe"[
  "MeshElementStyle" -> Directive[FaceForm[Green], EdgeForm[Red]]]]

And even simpler, you can use:

mesh = ToElementMesh[r, "BoundaryMeshGenerator" -> "OpenCascade"]
mesh["Wireframe"[
  "MeshElementStyle" -> Directive[FaceForm[Green], EdgeForm[Red]]]]

Here is a way to do what you want using DiscretizeGraphics as suggested by @user21. No need to call NDSolveFEM`

r = RegionDifference[Ball[{0, 0, 0}], Ball[{0, 0, 1}]]

rp = RegionPlot3D[r, PlotPoints -> 50]

Now we discretize the Graphics object with the following clever replacements that somehow works:

DiscretizeGraphics[Normal[rp /. {(PlotRange -> _) :> 
     PlotRange -> All, (Lighting -> _) :> Lighting -> Automatic}]]

To get even finer mesh, increase PlotPoints in RegionPlot (this will increase the time it takes to render the RegionPlot), then proceed as before. I used PlotPoints -> 100 for the image below: