Andie asks via Twitter:
We think of butter as primarily flavor and fat, and thus, butter should be superior to everything short of perhaps bacon grease whenever you need fat in a dish. However, there are a couple of peculiarities about butter that make it unsuitable for some cookies.
First, butter is not all fat; butter is about 20% water, and acts as an emulsion. This basically means that, when you're baking the cookies, the butter will give off steam. The extra water will cause a bit more gluten formation, so if you're going for a very crumbly cookie, then shortening might be a wise choice.
Second, and related to the first, while butter is solid at "room temperature," it has a very thin range of temperature where it is both pliable and not liquid. Too cold, and it is a block of solid matter so unyeilding that you can hardly push steel through it, much less cream some sugar into it. A few degrees warmer, and it's a puddle of liquid that couldn't hold bubbles for more than a part of a second.
Holding bubbles is a big job for the fat in a cookie. When you bake a cookie, those bubbles will expand in the oven, causing the cookie to rise a bit. If the butter melts before the cookie has set, then the bubbles are released, and the cookie falls. Of course, there's only so far that a cookie can fall, so it's not as disastrous as a cake falling, but cookie perfection is about subtlety.
The speed of the melting gets to the heart of what matters even more, which is the shape of the cookie at the end of the baking process. Butter melts at a lower temperature than shortening, which means it melts faster. The faster the melting, the more quickly the cookie will spread in the oven. If you want your cookie to hold its shape and rise up rather than being thin, then you need to use shortening. Shortening will hold its shape while the rest of the baking occurs. If you have some chemical leavening in the cookie as well, then the leavening will make the cookie expand, and as it's not going side to side, the other option is "up."