The difference between your viewport mesh and the render with the diffuse shader is certainly not due to a subdivision modifier. The volumetric render is corresponding to your viewport mesh.
It looks like you have a displacement (or perhaps bump/normal) added in your diffuse render, which is absent in your volumetric render (I cannot add any displacement or normals to volumetric absorption or scatter...).
When I try to reproduce what you see, I indeed lose the relief when it's applied as a bump map to the surface texture and I disconnect that surface texture. When I use the same bump map on the displacement input of the material output (and not on the normal input of the surface texture), it works as expected iff I select "experimental" in the render tab.