I've been using both of those since their earliest days. Both have come a long way, but in my opinion, their relative difference have remained remarkably consistent. In general, Lightroom is faster and easier to use, but DxO produces images with just a hair better quality in some specific areas.
The latest DxO noise reduction is remarkable. Combine that with the 6D's inherent outstanding low noise at high ISOs, and you enter a whole new world of low light image possibilities. I also find that if you are willing to put in the time, DxO can recover a little bit more detail than Lightroom (say in the corners of shots from fast lenses), but that will be visible mostly in large prints. DxO really improves images made with mediocre lenses via its correction processes (kit lenses suddenly become useful), but doesn't improve L lens images as much. Still better than Lightroom though.
On the other hand, you have to monkey with about a gazillion sliders to accomplish all that, and those who prefer a streamlined workflow always seem to lose patience with DxO. Lightroom default colors are usually a better starting point for me, for example.
Here's what it comes down to for me. I do all my DAM work in Lightroom, and process most of my shots there as well. But if I'm going to print something, especially something large, I almost always run it through DxO first. The only exceptions are shots from gear that DxO hasn't gotten around to analyzing (i.e. Sigma Merrill images).
