Why are savory crepes from some restaurants brown?
I suspect that you're lucky enough to have had the proper savoury pancake from some regions of France (e.g. Brittany). This isn't known as a crepe in French, but as a galette. This word also has other meanings, as a type of cake, tart or biscuit (British English)/ small crisp cookie (US English).
Wikipedia has an article discussing many types.  French Wikipedia lists many more. Here's a related picture from Wikipedia for comparison:



Breton galettes, i.e. those of the pancake variety, are made using buckwheat flour which itself has a mild savoury flavour, not exactly nutty but in that direction. They're a greyish brown rather than the golden colour of crepe. French names for buckwheat flour include farine de sarrasin and farine de blé noir, the latter meaning black wheat/grain/corn flour. Fillings are savoury (various cheeses, ham, etc.); occasionally nuts and honey are used together with savoury ingredients. Traditionally, galettes are paired with sparkling cider.

Galettes are not widely found outside France. In fact, even in France they are not easy to find outside Brittany, Normandy, the Loire region and the Vendee. I have been able to get them in the UK in the past. I've never seen them in the US (checking your profile), but haven't tried to find them either.