Larousse Gastronomique gives the procedure for clarifying butter as:
- Heat the butter gently until it has stopped bubbling and sizzling. By this stage, the water content has evaporated and a sediment forms in the base of the pan. Use a straining spoon to skim off any scum that forms on the surface.
- Pour or ladle the butter from the pan, leaving the sediment behind. The last of the butter can be strained through muslin (cheesecloth) if there is a significant amount left with the sediment.
Index - A Lexicon for the Indian Gourmet by Deepak Nirula defines ghee as "Clarified butter free of milk solids, with a nutty flavour."
So what, if anything, is the difference? From what I know, making ghee takes the clarification process a bit further, not just cooking out the water, but continuing to cook the butter until the milk solids in the bottom of the pan brown (but not burn) and the butter-oil takes on a deeper golden hue. The strained result can be stored at room temperature.
Removing all the water helps the keeping properties (no chance to mildew); removing the milk solids allows the clarified butter or ghee to be cooked at higher temperatures, where the solids would normally burn.
Hope this isn't TMI. :D