1. Tripods help you make more precise compositions. I'm not going to argue that tripods are freeing when it comes to composing, but I will say that once I've sighted what appears to be the ideal composition while handholding, I lock my camera to a tripod and then spend time making subtle compositional changes. I'm able to ensure that my horizon is level and that the edges of the frame fall exactly where I want them to. Because I use a geared tripod head (which I love for this very reason) I'm able to easily make very subtle compositional adjustments as I fine-tune my photographs.
2. What if the composition is perfect, but you want to refine elements within a scene? Tripods are ideal for that too, as they allow you to make subtle changes to elements within a shot—say the props in a still life, or the position of a portrait subject's hands or feet—without reinventing the photograph with every new exposure as you would if you were handholding. In this case the tripod becomes the literal foundation of your exposure so that the composition doesn't change from shot to shot, allowing you to easily see what other subtle changes you are making. Simple stuff, but oh so powerful.
3. Even if you're working with a fast shutter speed, tripods are still very helpful. I often find that when working in the studio with a pretty fast shutter speed of 1/125th, that if I'm not careful with my handholding, I'm sure to create a subtle bit of focus blur on a portrait subject's face. The chances of this blur are amplified if I'm using a long telephoto lens, but even with just a 50 or 70mm lens I've seen camera shake in evidence. I'm not alone when I say I've had more than one shot ruined because of this—and it's something that nine times out of 10 can be simply solved with a tripod.
4. If you're beginning to dabble with video, you'll soon learn a truism: video requires a tripod. (Or sticks, as they're called in the video world.) Those of us used to handholding our DSLRs for still photographs are in for a rude awakening when we see our handheld results when we turn our cameras to video. Sure, handheld video can be done well, and even shaky footage can have its place, but if you're planning to venture more into the video realm, a good tripod (with a video head) is invaluable.
5. Sometimes it can be very difficult to get your camera into the perfect position for a given photograph. Maybe the ideal vantage point is high above your head, or maybe it's just uncomfortably low to the ground. Either way, a tripod can alleviate some of the stress of getting to these tricky angles. Once you've reached that peculiar vantage point, the tripod obviously holds the camera there, so that you can make multiple attempts in creating the best photograph. And sometimes it would simply be impossible to handhold a camera in a position that a tripod enables—at least not with much accuracy or repeatability. I'd call that important, wouldn't you?
6. If you happen to be shooting straight down at a subject, there's no better way than with the help of a tripod. This way you can work to keep your shadow off of the subject, as well as helping to refine the angle of view and minimize distortion from this tricky-to-see angle. For more help in a situation like this, I like to tether my setup so I don't have to try to get my eye on top of the tripod-mounted camera. This way I can conveniently see the composition via the laptop, rather than struggling to get up to the viewfinder. It's not a vantage point I use all the time, but it is something I employ when needed. And it would be a lot harder to do it well without a tripod.
7. When I need to copy flat art, I mount it on the wall or a vertical flat and position my camera at a comfortable level on top of the tripod. Sure, I can point a camera at a wall pretty easily, but without the tripod I couldn't ensure that all of my straight lines remain nice and straight in the artwork I'm copying. Every little out-of-plumb movement that the camera makes is translated into significant distortion when it comes to copy work, so a tripod is mandatory if you want to do this kind of job well. Without it, it's just too hard to keep the subject plane parallel to the sensor plane—and that's the key to eliminating distortion.
8. Sometimes I make multiple exposures of a scene in order to composite them together into one superior finished image. Occasionally it's to increase the dynamic range of a particularly contrasty scene, but more often than not it's just so that I can modify scene elements or add bits of lighting here and there throughout the composition. A tripod is a necessity if you're making multiple exposure composites or panoramas. Not only should you use a tripod for this, but you'll want to leave your camera untouched as much as possible once the work begins, as there's nothing worse than kicking the tripod out of registration after many exposures.
9. If you want to create a repeatable photograph, there's no better way than to keep putting your camera in exactly the same place. And for that, a tripod is ideal. For instance, let's say you want to photograph a particular landscape across the four seasons. If you can mark the position of your tripod legs somehow, you're halfway there. The rest is just keeping track of the details of how you set the tripod legs and head. Note that you had the tripod at full extension, for instance, with the head rotated 35 degrees. Don't forget to note the details of the camera and lens, too. Being able to return the camera to the same exact position is no easy feat, but a tripod makes it much more doable.
10. Along those same lines, somewhere between a multiple exposure composite and returning to the same scene again and again is the creation of separate photographs in separate places to be composited into a finished photograph. It's about recreating the same perspective at a later date, in a different place. Maybe you want to photograph a background here today, and a person there tomorrow, and a prop or two somewhere else at some other time. In these situations, it's imperative not only that you light these elements the same way, but that you position the camera the same way as well. Again, knowing the height of the camera, the focal length of the lens and the exact angle at which you've got the tripod head set allows you to recreate the same perspective in another shot. Thankfully, you can track the position and angle of the tripod head by noting the hash-mark indicators on many tripod heads. Failing that, you can use a protractor to measure the angle at which the camera is tilted. This way individual elements photographed in different places and at different times will all share the same perspective, and if you match the lighting, too, you'll make for a much more seamless and more effective composite in the end.
If you have your own favorite reasons to use a tripod, please share them in the comments.