The main advantage is cost. Historically, there were two main types of imaging sensors, CMOS and CCD. CMOS was historically used in digital cameras (because of the higher quality images it can capture, the lower cost, and the fact that digital cameras only have to capture a single moment).
CCDs on the other hand were historically used in video cameras, with one CCD per color (3CCD) to increase the quality. They have been significantly more expensive and generally capable of lower resolutions, but had the advantage of capturing the entire frame at the same time.
Most CMOS sensors scan line by line rather than capturing a fixed image at a time, they exhibit an artifact known as a rolling shutter. This appears as a distortion when the camera is moving (particularly panning) due to the change of orientation mid-frame. Previously, the sample rates for CMOS sensors were too slow to allow their use for video, but as the technology has improved, it is now possible to sample quickly enough that, while still present, rolling shutter is not nearly the problem it was even a few years ago. Now there are even CMOS sensors capable of offering a global shutter and they have all but supplanted the old CCD based cameras, with even major motion pictures being shot on CMOS sensors with global shutters.
The advances in CMOS technology have allowed for CMOS sensors to be used in professional level video, bringing with them the low cost, high resolution and high quality that had previously not been possible. Global shutter CMOS still costs more than rolling shutter relative to other features of the sensor, so rolling shutter is still present in many cameras (even fairly high end cinema cameras) due to the cost savings and the fact that higher sampling speed has greatly reduced the impact of rolling shutters.
Another large advantage is size. DSLRs tend to have less processing capability to work with the large amounts of data involved in things like 4k video, so they are more limited in capability, but they also are much smaller and more portable. This is probably most visible in comparing the Black Magic Production Camera and the Black Magic Ursa Mini 4k. Both use the same 4k sensor, however the size of the BMPC limits it's ability to work with high frame rate video as well as some other features of the camera. The Ursa Mini is a much larger form factor that currently costs the exact same amount, but is able to fit more features in due to the extra size and hardware that can be included in the larger form factor.
