I regularly use both the 5D and the 1D.
The good news is that they are both great!
As has been mentioned, the 1D has better AF -- use the two side-by-side and this will be evident.
Another feature the 1D has is Highlight Tone Priority. I've tested this, and it applies to RAW files. It is similar to underexposing by a 1 stop lower ISO then boosting the low and mid tones digitally while keeping the highlight detail and can be a timesaver and even an image saver when highlight details are important.
The 14 bit image is meaningful, but in practice I'm not sure how much of an edge it gives. But that's just me.
Some here claim that the Mk III has better high ISO/low noise performance. I haven't seen the tests that demonstrate that, but I've run tests of my own and have seen more chroma noise in shadows with the Mk III, consistent with the higher pixel density/smaller photon sites you have with the Mk III, so I would respectfully disagree with those assertions. However, the default chroma noise reduction applied by Lightroom makes the difference negligible.
I don't hesitate to use either camera. Different situations will prompt me to grab one body or the other. In general, I don't see the Mk III images standing out above the 5D, so when the 5D with ff and its compact body suit the situation, I don't feel like I'm compromising when I use it.
If I had to choose just one of the two bodies, I'd probably choose the Mk III because I do a lot of wildlife photography and the Mk III has an edge there with its AF capabilities, but if I did exclusively landscapes, portraits, weddings, or street photography I very well might choose the 5D.
Fortunately, I don't have to make that choice! In fact, I also have a 30D which makes great images, and I loan that to my son so he can be a "second" when we go out!