They´re both excellent cameras. I´d say if price is no matter, the Mark III wins for achieving a beautiful look with lesser effort.
The one thing that really stands out on the GH3 (I don´t think the Mark III has it?) is the extended tele mode. Unbelievable useful. put on the superb panasonic 12-35 and with two tabs on the monitor you got 28.8-84mm from the same lens by maintaining image quality (though you don´t get the lens characteristic of a true tele of course)... saved me lots of times already. The GH3 is very fast to use if you know it. Bevor recording I switch to autofocus, camera does the work, back to manuel mode, record. takes no more than two seconds and you´re always there.
I´d say the Mark III´s advantage is Magic Lantern support, the full frame wiith it´s shallow depth of field and the EF mount. I know you can put on anything on the GH3 but who really wants a full frame 24mm lens instead of the voigtländer or the slr magic? So once you buy yourself into mft lenses you can only hope that there will always be some outstanding mft cameras... while EF lenses, well... you can use them on any sensor size, and any brand.
The GH3 is cheap but needs expensive lenses to really shine while the Mark III is expensive but can achieve a more pleasing result out of cheap lenses... but then again... better burn money on lenses than bodies, their worth will always be high.
In lowlight the GH3 will need fast lenses, for I wouldn´t go further than 800 ISO, 1600 is ok too if you got something like neat video. The noise looks acutally very nice, but it´s not for every project... The canon compensates its inability to manage focus on fast lenses with higher ISO and higher f#.
so... basically, you´re good whatever you take :)