You've actually stated the answer in your question: it's the wider field of view that's yielded when you mount a smaller-format full-frame fisheye onto a larger format and shave the hood so that it no longer impinges on the frame.
There's less distortion, better sharpness and chromatic aberration control on a diagonal fisheye than with a circular. And sometimes they're cheaper (e.g., a Samyang manual fisheye lens can be $250 vs. $600 for said Sigma 8mm).
And the view coverage is crucial when shooting 360x180 spherical panos, because it can cut the required number of member images down to something easily handholdable.
With my Samyang 7.5mm for micro four-thirds on my DMC-G3, I only have 135° coverage. To cover the sphere, I have to rotate the camera in yaw at 60° intervals to take six shots around, AND a zenith (straight up) and nadir (straight down). If I were better at judging slight angles, I could tilt the camera up 15° for the around shots, and then only need to take a nadir. But I kind of suck at judging the 60° accurately (I've shot a lot of handheld panos with holes with this combo). Generally, I pull out the tripod and panohead instead of the plumbline.
By contrast, a NEX or Fuji X user who buys the mft 7.5 and an adapter ring, and then removes or shaves the hood, can cover the same scene with four shots rotated around at 90° intervals, just as I do with the Sigma 8mm circular on a 5DMkII. Much easier to train yourself to do.