The jpeg definitely looks like it has compression posterization (the banding in the sky). Thats usually caused by high levels of compression with smooth gradients. You'll need to compress very little or not at all to eliminate the banding in the jpeg.
As for the screenshot, I'm not exactly sure what I'm looking at. It definitely looks better than the jpeg, so I doubt that is the reason for the banding. LR 4.1 is a release candidate, and its support for 5D III RAW's is preliminary. I would double-check the original CR2 files in Canon DPP to make sure the issue isn't "baked in". If everything looks fine in DPP, then the issue is LR4's support for 5D III RAW, and you should give Adobe a big, long, loud holler (LR4 has a LOT of issues, not the least of which are some pretty severe performance issues, so they may need an extra kick in the groin to get their act in gear and make LR4 what people have paid for it to be...including 5D III support!)