Tripods designed for still cameras and those meant for video, while similar, are very different beasts. They both have three legs, of course. They're both designed to provide a steady platform for your camera. But there's a critical difference between the two.
It's in the head of the tripod, the part where you mount your camera. Video tripod heads are called "fluid" heads. They're designed specifically to allow you to pan the camera from side to side and tilt it up and down smoothly while taping. Try these moves using a video camera on a tripod designed for a still camera and you're in for nothing but herky jerky movement.
If you don't want to spring for a video tripod, check first to make sure your still camera tripod will hold the weight of your video camera (video cameras tend to be heavier than still cameras). Then set up your shot, start recording and don't touch the tripod. It's okay to readjust the camera position between shots, just don't do it while you're recording. However, one of the attractions of video is the movement it allows. You'll find that camera moves, especially when taping family photos, add to the quality of your video. So even if you don't think you'll use it often, I encourage you to at least buy an inexpensive video tripod. I just bought a very nice Manfrotto for about $250. But you can find video tripods less expensive than that.