The above summarizes it pretty well, but also:
Heavy moire and aliasing with patterns and sharp detail (especially with 60D)
No built in ND filters (can get away with screw on filters, but pain for broadcast and fast changing light)
Recording time is also limited because the sensor/chip overheats and a lot more than dedicated video camera.
The 60D doesn't allow for external recording (at least not clean HDMI out)
The Black Magic is special because of it's outputs, quality sensor and high dynamic range for the price.
It can shoot video in very usable compressions/codecs for better editing and can even shoot RAW.
The higher dynamic range means it's closer to a cinematic look, and allows for greater flexibility in exposing and editing.
The "Jello Effect" is called Rolling Shutter and is a common problem with cameras using CMOS sensors.
As when the image is read from top to bottom of the sensor, fast action will not be recorded all at once uniformly across the sensor so movement of the camera and subjects are all affected.
No it doesn't take a genius to fix this, just some cameras handle this better, but there's no way to get rid of it.
Software can help with fixing but generally doesn't look good.
The Black Magic Production camera that's supposed to come out in July has a global shutter, which rids the rolling shutter effect completely.
Focusing is done manually for greater control, as AF in video will often not focus to where you want it to.
And for greater creative use it's better to follow focus when there's shallow depth of field.
An advantage that the 60D and other DSLR styled cameras have though is the size and weight.
The flip screen especially helps with shooting from odd angles, and it can be mounted on various rigs.
I use a GH3 which has a similar form factor, and it's miles ahead in every aspect for video, all the 60D has
is a larger sensor, but it's not that much bigger anyways. Also, the GH3 has no mirror, so virtually any lens can be
adapted to it, which can be useful.
And in other news, the Magic Lantern team that unlocked RAW video in certain cameras, especially the 5D Mark III is exciting many people, and if image quality is all that matters, then that camera with the eventual release of the firmware hack (and a boat load of expensive 1000X cards) is what to get for just about the best in video quality.