There are times when you’ll want to use a tripod to compensate for blurriness caused by camera shake: when taking pictures in low light, where the shutter speed will slow down enough to potentially make drag lines, and when using a long zoom, where distant objects are susceptible to blur. In each case, the tripod will settle the image and let you forget about shaking the image and focus on capturing what’s in your mind’s eye.
Photography of some weather subjects requires a tripod-mounted camera: lightning, aurora, noctilucent clouds, mesospheric clouds, sunsets and twilight colors, the sun and moon, green flash, zodiacal light, star trails, macro photography and all nighttime photography can only be done well with some sort of tripod. After the camera and lenses, the tripod is I think the most important piece of equipment to have if you want to photograph the weather (or nature in general).
Happy Shooting!