I have a 6D, which I've been using in low light situations. In recent months I've found myself frequently exploring the limits I can go and still come up with what I consider successful photos. I've very happy with the results I've been getting up to ISO6400 or so.
After downsizing 22mp 5d3 -> 20mp 6d and adjusting for slight sharpness differences it's a wash, this is the same sensor generation. However the 6d has significantly less banding, i.e. higher dynamic range which *does* make a difference if your shots require it - it even tops 1dx on low iso. You can boost the dr on 6d even higher with Magic Lantern, nearly up to 15ev.
Having said that, no nice noise performance makes up for dodgy metering or getting out of focus shots with the ancient 6d af unless you're mf'ing (like with ML's focus peaking).
Even what I've shot at 12800 pleasantly surprises me.
ISO 12800 is just a digital push (on 6d, actually even 6400 is) - no need to use it unless you shoot jpeg or do video, otherwise underexpose a lower iso and keep the higher dynamic range of the lower setting.
The main positive about the 6D is the centre point is rated to -3ev whereas the 5DMKIII is -2EV.
Well, yes, it's "nice to have" esp. with slower lenses like f4+. *But* af'ing @-3lv is so slow that you could also just switch to live view and contrast af or mf with a boosted lv (ML again) which works far below -3lv.