IMO the 5D3 has the best AF system for BIF of any Canon body I've ever used, including my Mark IV. My recent 5D3 Burrowing Owl series confirms this impression. If only Canon would put the new AF system into a 1.3 crop body. That would be the ultimate BIF body!

While off topic I was very happy with the multiple frame "silent mode" the camera shoots in. I rented the camera and shot nesting eagles and falcons and found the quiet shutter to be a great feature. As I typically shoot skittish wildlife, I could see owning this camera for that feature alone.

I find the 5DIII is on par, and perhaps a bit better than the IV for BIF tracking. As mentioned, it is a bit slower in raw drive speed. What I like most however, is how much I can customize the AF modes. I was shooting black terns (extremely erratic flyers) the other day and while my comparisons were by no means scientific, I felt my keeper rate with the 5 was substantially better.

As a result, the EOS-1D X supports EOS iTR AF (Intelligent Tracking and Recognition) that can apply face and color detection data from the 100,000 pixel sensor for improved subject tracking when the camera is set for automatic focusing point selection.
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The "face detection" difference between the two Canon cameras does not improve the birds AF at all, but the "color detection" that several Nikon and only the 1DX have seems important to fast AF. Sadly the 5D-III does not have it. That can make a difference in BIF.