"Somewhere else" names a specific, though perhaps unknown, place. By contrast, "elsewhere" is nonspecific. Here is an example to show that "somewhere else" and "elsewhere" differ slightly:
"Does Tom have friends in Chicago?"
"No, his friends live elsewhere." (His friends are not in Chicago. They may be in many places.)
"No, his friends live somewhere else." (His friends live in a different city, but all in the same city)