In the US, when mulatto was in more widespread use, the term was used to describe Turks as well as persons of white/south Asian and white/ Native American backgrounds. So it is not necessarily more specific, in practice. I think the best reason not to use the term when in North American circles is that those it refers to generally strongly dislike it because of its poisoned historical past, which created an implicit comparison between mixed people and sterile mules, or more generously, strangers in their own land. Imagine if the colloquial term for Americans of Irish/German ancestry was "banana slug" or "interloper".