Before getting the 1DX I was shooting with a 1DIV for theater & concerts and a 1DsIII for actor/dancer/singer headshots.
I absolutely love the IQ of the 1DsIII! The build quality is what you'd expect from a 1 series...it's a tank. Functionality wise, it's never shot an error code at me. I use live view for interior architecture shots and never had a problem. But I've never shot tethered.
As others have indicated, it doesn't have the high iso capabilities of more recent bodies.
It is slower to focus. When I'm shooting headshots I'm using the 85 1.2L II 85% of the time. This lens is known for its slow focus, so it doesn't bring out the best of the 1DsIII, but I'm not in need of instant focus in this situation. The IQ of that body/lens combination is special. I've never been irritated with the focus speed.
Also, it has fewer focus points.
I sent mine to CPS for a clean & check several months ago and, without asking, they replaced the shutter. I used it one more time and it has been sitting since. I was reluctant to give it up because I wasn't sure if the 1DX could match its IQ for headshots. I still think that the 1DsIII might be better in that regard, but that opinion is anecdotal and could be me overly scrutinizing the 1DX. I recently did an actresses headshots for the 2nd time. Her first shots, 18 months ago, were w the 1DsIII and this time I used the 1DX. She loves them just the same. I don't think you could tell the difference if I handed you both prints.