"Shadow function" gives you an alternative way of defining which parts of the atmosphere are in shadow. This can be really useful if you need certain areas of the atmosphere to be in shadow but you don't want to enable "received shadows from surfaces" because it can be slow. For example, if you want everything below a certain altitude to be in shadow, you can do this with functions or shaders. For the Paramount 100 Years ident, I used a combination of altitude-based functions and Simple Shape Shaders to create the shadow of the low-lying hills and the main mountain. It was fiddly to set up, but well worth it for the render time saved.
Matt