Sometimes, you find that the open end of a wrench spreads under heavy load (which oft leads to rounded corners.) You would naturally reach for a box end wrench, but sometimes, there will be something in the way that prevents you from putting the box end wrench over the nut. Brake lines are a classic example of this. The hard line prevents you from using a box end wrench, but an open end wrench will often round the nut. Couple that with lines using softer nuts (to hold the flared end), and you have a recipe for frustration.
Enter the flare nut wrench (so called because you use it on nuts that hold flared lines.) It's also called a line wrench (because, well, you use it on lines.) There's a cut out so you can get the wrench around the line, and then down onto the nut. But the wrench also has five corners, which grip better than an open ended wrench.
You might also find them pretty handy on brake bleeder valves. You can generally use a box end wrench to minimize rounded corners, but some calipers are designed so that a box end is almost impossible to slip on. A flare nut wrench often has a better chance in those circumstances.
Some people will recommend skipping flare nut wrenches, and buying flare nut crowfoot(s) instead. The idea is that both are limited use tools, so if you can combine them, you can save some space/weight in your tool box. This is what I chose to do. I was a little hesitant because if you're using crowfoot wrenches, you're already tight on space, and finding the space to slip the flare nut crowfoot over a nut instead of an open end crowfoot onto a nut might be tough. Thankfully, I haven't run into that particular problem yet.