Why can't you just put a frame of fisheye video on the dome?
Well, on some very small personal domes, you can. But for most domes designed to hold more than 2 or 3 people, especially fulldome theaters designed to hold a full audience, the domed screen simply requires a lot more pixels than a video camera can create, even more than a hi-def camera can create, in a single framed shot. DomeFest, for instance, is requesting dome masters measuring 3200 x 3200 pixels.
The image to the right shows the relative sizes of a 3200 x 3200 dome master, a standard frame of video and a frame of high definition video. You can clearly see that even the full 1080 hi-def frame is less than one third as big as it needs to be for fulldome.
Often, the first thought is to try to stitch several video images together. Just remember that there is also a drop off in color at the edges of a frame of video. So you will not get a seamless image by simply butting video frames together.
There are people working on ways to do this (Micoy is one), but unless you have access to some very high end video equipment, you'll have to get creative here.