Is brown butter the same as ghee?

BB is basically butter cooked into ghee and used along with the browned milk solids, GHee is filtered to remove the crusty bits

On a side note, the uses for brown butter and ghee are very different. Brown butter is used primarily to add flavor to a dish, and it is often used as a finishing element (or, it can be used in baked goods for another layer of deliciousness). Ghee is often used in place of olive or vegetable oil when sautéing. One of the reasons for ghee's prominence in Indian cuisine is because removing the milk solids extends the shelf life of butter (which is definitely an advantage for those without refrigeration in hot climes). Also, dairy products hold something of an esteemed place in Indian cooking--to oversimplify a bit, it's part of the life-giving force of the sacred cow.

Ghee is made by browning the butter, then removing the flavorful fat, leaving the solids behind. French beurre noisette (brown butter) leaves the solids in.

In this case, you can trust your eyes. Brown butter contains the browned milk solids, ghee does not.