Here's something I posted a while ago explaining the differences. This is more geared towards to Xf300 (canon's flagship small-sensor video camera), but it's a similar story to most small sensor video cams.
*I'm assuming by 'professional video camera' you are talking about small sensor ones, not the pro cine ones. Basically, I'm assuming you talking about the type news boradcasters use.
"The 3-chip XF300 will be super sharp, but remember, It's not a replacement for your DSLR. It's a small sensor video cam that's more geared for news broadcast. You will not achieve the 'dslr' look with it. The XF300 has a 1/3'' sensor which equates to a 7.21x crop. I've used 1/3'' videocams as well and I vastly prefer the 5d3 over them. Even though the 5d3 is softer, the image looks so, so much nicer. With the XF300, you'll have to deal with abysmal (compared to the 5d3) low light performance and you won't be able to get any blurry background goodness as easily. What you do get with the 300 is a razor sharp HD image, broadcast ready codec (50mbs 4:2:2 MXF) and pro video features (peaking, zebras, XLR inputs etc). But the two CANNOT compete wen it comes to the images. They're two COMPLETELY different cameras for different purposes. XF300 is more suited to news gathering, ENG work, whereas the mark3 is geared better towards cinematic work. (you can use them vice versa, of course...but in general, you get my point...it's harder to use the 5d for ENG and harder to get the same filmic look with the 300...not impossible, though)"
Bottom line:
small sensor video cam:
-Good lens reach (usually 30-600+ mm)
-terrible low light performance (compared to a 5d3, for example)
-Really nice, sharp, full HD image
-full controls on the body for easy access
-good internal codec
-proper audio inputs
-proper video features
DSLRs:
-Great low light performance (in general)
-image slighty soft dye to downscailling. (the FULL HD tag is a lie, its HD, but hardly full 1080)
-really nice cinematic look/DoF
-no XLRs for audio
-usually* pretty poor internal codec
By "insanely fast lenses for video work"
Assuming small sensor video cams, yes their lenses generally start at f/1.6 - f/1.8. The high end models are constant f/1.6 while the cheaper ones (sub 10k) are variable and end at f/3.6-ish.
The F/1.6, is, however, offset by the horrible high ISO noise performance on the tiny sensors.
*** now, if you're talking about the cine pro video cams (alexa, c300, fs100 etc) with large sensors, then there's no contest. They beat DSLRs hands down in almost every category when it comes to video. They also costs many times more.