In general relativity, is there a difference between matter-energy and spacetime? This is the third time I try to formulate a question in a proper format so as to understand what is going on with gravitational waves at a physical level.
I am told that a spacetime oscillation and/or a gravitational wave in the $z\text{-}$direction can move particles in the perpendicular plane. Actually whatever the polarization, what is important here is that those waves move things.
Now, if what makes something move matter or light (referred in the question as "matter-energy") is called "spacetime" and thus there is no difference between the supposed container (spacetime) and the contents (energy, matter, light), especially since gravitational waves carry energy, then what is the difference of nature between the two, or actually is there any at all?
 A: Yes, there is a difference. You can have spacetime without matter or radiation in it, but you cannot have matter or radiation that is not in spacetime.
Matter and radiation exist in, and move through, spacetime. But spacetime has its own dynamics. Its geometry is determined by matter and radiation, and that geometry affects how natter and radiation move.
In currently accepted physics, reality consists of two fundamental and separate things: spacetime, and quantum fields in spacetime that describe matter and radiation.
A: As you point out, in terms of GR spacetime is a dynamic participant in the physics taking place. Spacetime that is unstressed exists in a geometrically flat state. Presence of stress and/or energy density induces a state away from that: spacetime curvature. That stressed state is itself a source of gravity.
There is a lecture by Einstein, written around 1921, written for the occasion of accepting a guest lectureship at the University of Leiden. The lecture is titled: 'Ether and the theory of relativity'.
Einstein points out that GR attributes physical properties to spacetime, a state of affairs that parallels the way physical properties were attributed to the Ether. At the same time Einstein stressed the following characteristic of GR spacetime: it does not have parts that can be tracked through time. 
Here is how I understand that: for all other physical phenomena we have a way of assigning a velocity vector as a part of how it is described; you can track it though time. But spacetime does not have parts that can be tracked through time.  
A: I'm taking the unusual step of posting a second answer, in order to reply to a comment. G. Smith has raised an important point, one that merits thorough discussion.
I argued in a comment to an answer by G. Smith that the assertion that: 

You can have spacetime without matter or radiation in it

is too bold: I argued that we cannot exclude the possibility that in fact a universe devoid of matter or radiation cannot exist.
I agree with G. Smith that in this forum (physics.stackexchange) for most questions we can treat our current theories of physics as simply correct.
If a successor to GR and QFT is formulated (presumably unifying the two) then as we know this successor-theory must reproduce all of the established features of its predecessors. For instance, since we have strong evidence for the existence of gravitational waves we judge that any successor to GR must include the existence of gravitational waves. So when answering a question about gravitational waves we have the assurance that even if GR is eventually replaced statements based on GR will still be good.
However, in this particular case the concept is very fundamental: without any matter or radiation in it, can spacetime exist? Unlike the case of gravitational waves this is a question that is not open to experiment. 
Using the terminology of the orginal Question: over time the physics community has seen the distinction between 'container'  and 'contents' becoming less defined. This suggests that the two are quite interdependent, possibly more so than we currently know of.
At that deep level, there is room for a successor theory to deviate from its predecessor.
General remark:
To allow for the possibility that our current theories are not exhaustively correct is part of mainstream physics. For example, particle physicists are doing work to achieve progress within the realm of our current theories, while at the same time they are eager to find clues to physics beyond the Standard Model. Those two objectives can and do co-exist.
A: User WillO wrote the following comment:

why isn't Minkowski space a counterexample to the expectation you
  attribute to Einstein?

I will write a temporary answer here, as it is too large to fit into a comment. WillO, if you submit your question as a new question I will move this answer to that new page.
About Einstein's expectation:
Einstein had a hunch that the existence of inertia (in the universe) is due to all of the presence of inertial mass in the Universe, in the sense that perhaps all of the inertial mass in the Universe combined is what gives rise to the existence of inertia in the Universe. If that is the case then a Universe without any inertial mass in it would not have inertia. It is my understanding that the solution called 'De Sitter space' is at odds with that hunch.
About the difference between Minkowski spacetime and GR spacetime:
John Wheeler coined the following summary: 'Matter is telling space how to curve, space is telling matter how to move' Of course that was not intended as an accurate description; the purpose was to focus on a specific feature. In terms of GR spacetime is a dynamic participant in the physics taking place. Spacetime is affected by (lumps of) inertial mass, taking on curvature. 
Where GR spacetime is a dynamic entity, Minkowski spacetime is a static entity. Minkowski spacetime does affect the physical processes occurring in it, as illustrated by the Twin scenario, but Minkowski spacetime itself is not subject to any form of change.
As we know, universal Minkowski spacetime is incompatible with GR.
We have the succession of three theories of space and time  


*

*Newtonian space and time  

*Special Relativity  

*General Relativity  
At both stages of that succession sequence the old theory is a limiting case of the theory that superceded it.
Special relativity superseded Newtonian space and time. Limiting case: in the realm of non-relativistic velocities the mathematical expressions simplify to newtonian dynamics
General Relativity superseded Special Relativity. Limiting case: in the realm of negligable curvature GR spacetime is observationally indistinguishable from Minkowski spacetime.
In terms of GR Minkowski spacetime is an obsolete concept, part of the previous theory of space and time.
A: The way your question is phrased might be applied to something more familiar, namely electromagnetism. There you have an electromagnetic field which exerts a force on charged matter. But it does not follow that an electromagnetic field is the same sort of thing as electric charge. 
This analogy with electromagnetism works quite fully when you look at the mathematical equations of classical (i.e. not quantum) general relativity. On the left you have partial derivatives of a quantity $g_{ab}$ which is a measure of spacetime, and on the right you have a source term $T_{ab}$ which is a measure of the matter and stuff which can gravitate. The equation is not saying that the one is the same as the other, it is saying that the amount of spatial and temporal variation of the one is determined by the local density of the other. 
I think your question does alert us to the notion that spacetime should not be regarded as a mere nothing, a mere way of arranging labels for position and time. It is rather a very specific type of arena in which space and time labels have to satisfy certain rules of consistency (e.g. it is pseudo-Riemannian, not some other type of manifold). Spacetime or its tensorial gradient (take your pick) can be correctly regarded as a type of field.  
