Heat a very good quality unsalted butter until it smells of hazelnuts. The color will be golden brown, but the butter hasn't been burned (yet). Basically you use your nose when making beurre noisette.
You can keep beurre noisette in the fridge for 2 - 3 weeks.
Clarified butter is different in that you heat the butter on a slower fire. This will separate the milk solids (proteins such as whey and casein) from the fat (the water in the butter evaporates). Commercial clarified butter can be made by direct evaporation, decantation and centrifugation.
To make beurre noisette you first go through the process of making clarified butter. Then increase the heat and toast the milk solids brown.
In India clarified butter is known as ghee. There are many cultures which use some type of clarified butter. Sometimes, like in Ethiopia, the clarified butter is spiced with cumin, coriander, turmeric, cardamom, cinnamon, or nutmeg. In Morocco the spiced ghee is aged for months and sometimes, years. This is called 'smen'.