I bought my 70-200 f/4 VR III about three weeks ago. I didn't have a lot of opportunity to shoot with it, but the first impressions are great.
I compared it with 70-300 f/4,5-5,6 VR, good old faithful 80-200 f/2,8 ED, and with 70-200 f/2,8 VR II.
It won the comparison with 70-300 f/4,5-5,6 VR hands down, 80-200 f/2,8 was very close in the range between 80-135 mm, but the real competitor was 70-200 f/2,8 VR II. I made some quick shots in relatively low light and longer exposure times and found both lenses to be pretty close, the "big brother" was maybe a tiny bit sharper in the center, but 70-200 f/4 VR III was much better in the corners.
What impressed me the most was the new Vibration Reduction. You can see the difference already in the finder, the picture is visibly more stable and the shots with 1/60 s at 200 mm were perfectly sharp, better than with 70-200 f/2,8 VR II.
I am very pleased with the lens. I need it for travels where 70-200 f/2,8 VR II is simply too heavy.