What are the advantages camcorders have over zoom digital cameras?
The video capabilities of even the best digital cameras are but a small fraction of these of video cameras. The gap is smaller now but only marginally so.

Where digital cameras compare very well is in terms of image quality, particularly when it comes to low light shooting. For almost everything else, video cameras trump digital cameras.

Camcorders have sophisticated and detailed control over audio recording. They can set levels and recording angle which almost no digital camera can do. You can plug an external audio source on some still cameras but that is about it.

The optical zoom of camcorders is specially designed for video. It moves silently and smoothly to give a natural look. Camcorder lenses are built with a continuous variable apertures so that the brightness of images can vary smoothly rather than in steps (only a handfull of SLD lenses can do that). The focus system works equally smoothly and quietly while most digital cameras record AF noise in the audio track.

Camcorders have tons of features which are meaningful to video. Fading, transitions and even video effects are the norm. They can also record for much longer both continuously and in sequence than still cameras. You can get up to a 128GB SDXC card for a still camera but it is costly while a camcorder can easily have a 200GB HDD built in.