Some points to modulate in order to achieve your desired result (I assume the bottom and background is one sheet of paper curved up) :
0) you do shoot manual, with the same exposure, right ?
1) put your lights + object further from the "background" part of the white paper -> the background will become darker (curve the paper less to obtain a nice gradient - and this will also work with darker objects), the subject will remain the same : less blending
2) lower the camera (in order to emphasize 1) : subject is even more detached from grey background
3) "focus" your light sources more on the subject than the background... you didn"t specify what kind, but it looks like lightboxes : try changing the orientation of the sources towards you, or putting opaque masking on the furthermost part of the lightbox.
And if you can opt for a slightly different render :
4) use an additional light (very soft bowl or small box) placed above your subject, oriented towards it (example).
After some thought, here are my ideas regarding the consistency issue...
On the first monitor where I saw your photos the difference was barely noticeable, but on a better one it is slightly more visible, I even wondered whether they were shot with the same exact exposure. And I don't know whether they were taken at "exactly" the same time or not.
Some leads you could check :
rig consistency : ensure that your equipment (lighting, background paper, camera) are all in the same exact position. For example changing lighting distance from 1m to 1.4m would halve the light amount (this is purposely exaggerated, but a smaller variation could go unnoticed while shooting).
lighting consistency : assuming your lighting equipment itself has reasonably good quality, it should provide relatively good reproductibility (if cobra flash, use manual setting, not TTL-ish). But there might be other sources of light that can interfere, like fluo or tungsten bulbs or windows. While not sufficient to light the scene, a sudden change of sunlight (cloud...) could explain the difference, or the fluctuations due to the nature of AC mains.
camera settings consistency : you shoot manual, but there are other parameters you need to anchor, like the white balance (for example, select the one matching your lighting technology) and mind the possible other alterations produced in-camera (here, D-Lighting)
image processing software consistency : import and process the images with the same settings exactly (regarding exposure, brightness, contrast, colour...)