DSLR's have an advantage in that you have a large selection of lenses and the lens can change the feel of the image. For example, some lenses allow you to create a fairly narrow region where things are in sharp focus (we call this "depth of field") and deliberately blur the background -- you see this in cinematography all the time where the camera focuses on the person speaking and the rest of the shot is deliberately blurry. Of course this presumes you're going to buy more lenses and that can be fairly expensive.
Camcorders tend to come with one fixed lens (you can't change it) and it's usually a fairly forgiving lens.
DSLRs are usually NOT intended for one long continuous take... usually you're taking a few minutes of video at a time whereas on a camcorder you could tell it to record for an hour and as long as the battery does die it'll just keep shooting.
But camcorders have several ease-of-use features designed specifically for video... continuous auto-focus, excellent subject tracking, and they don't build up as much heat as a DSLR and part of that is because they are low-resolution cameras. Even 1080p hi-def video is only 1920x1080 and that works out to about 2 megapixels. A DSLR will give you 16, 18, 20 or more megapixels which video can't use. So the DSLR is using a sensor intended for high resolution imaging but reducing the resolution when it saves the output. It turns out the sensor can get hot and since it's primary purpose is for "still" imagery, it doesn't have heat sinks to dump the heat (which is why they're best for fairly short duration clips... e.g. 5 minutes is no problem but typically not more than 10 minutes in a clip and then you let the camera rest.)
MANY DSLR cameras cannot do "continuous" autofocus when shooting video. Some newer models can but it's not safe to presume a camera can do it. Pro video people don't really care about it since they tend to not use auto-focus.
Canon's new 70D has a new feature which allows continuous phase-detect auto-focus to work on the sensor (this is something previously not possible in a DSLR). Phase-detect auto-focus is faster, more accurate, and doesn't require "focus hunt" (when you see the camera overcompensate focus and then have to move back to get things in focus). Their new "STM" (stepper motor) lenses are virtually silent to the internal microphone (although external mics such as a Rode VideoMic will typically have much higher audio quality.) The 70D has several features that tilt it in favor of shooting video. It's not perfect and it does have it warts... but overall it's one of the best.
A camcorder is going to be easier to use -- the whole thing is optimized for video.
A DSLR is going to create the potential of a higher quality look to the video -- a bit more of a cinematic feel. But as it's not really optimized for video it takes a bit of learning the nuances. It also presumes you'll shoot lots of short clips and then put them together in software (ever notice how often the scene changes in a movie or TV show... it's typically hard to count more than 10 seconds before the scene changes to a new angle... rather than the one-big-long-continuous-take look of amateur video.)
One last thing to keep in mind is that you can do fantastic still photography with the DSLR and the right lighting. Good "light" requires good "shadows". If the light is located near the camera lens then the body contours wont really create shadow and that makes the body appear "flat". But if the light is off to the side it will put highlights on one side and shadow on the other which really creates some dimensionality to body contours. The key is it needs to be diffuse light (which means it appears to originate from a fairly broad area rather than a pinpoint origin). This causes a nice transition from light to shadow rather than a harshly defined line between light and shadow.
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