Q:

What is the difference (if any) between _mien_ and _demeanor_?

Mien is a word I don't encounter much, but when I do it always seems like it would have been interchangeable with demeanor. Dictionary definitions don't really clarify much (they still seem interchangeable).
Both seem to refer to someone's outward characteristics (look and behaviour) as a reflection of that person's likely or suspected inner characteristics (mood and temperament).
Are they totally interchangeable, or do they have a subtle difference in meaning or usage? Are there any contexts where you might use one, but the other would be inappropriate?

A:

Mien and demeanor are close, but the latter is somewhat of a subset of the former. Mien typically includes a sense of carriage or posture, even a trace of stiffness. One would not say of someone:

He had the mien of a hobo

because hobos aren't typically associated with upright bearing. From Merriam-Webster:

Like its synonyms bearing and demeanor, mien means the outward manifestation of personality or attitude. Bearing is the most general, but now usually implies characteristic posture, as in "a woman of regal bearing." Demeanor suggests attitude expressed through outward behavior in the presence of others; for example, "the manager's professional demeanor." Mien is a somewhat literary term referring to both bearing and demeanor. "A mien of supreme self-satisfaction" is a typical use. Mien and demeanor are also linked through etymology...from Latin mener ("to lead"). 

Here's an example from Toru Dutt's sonnet "The Lotus":

    The rose can never tower
  Like the pale lily with her Juno mien

The connotation is that the lily is taller, more erect, more regal than the rose. 

