So far it is the moderator which is easier to separate. My association goes that it is the person who looks after a forum, i.e. checks relevance and quality. Facilitator in turn, would be the one who makes the discussion go on towards some desired end. And a teacher (I am aware of what Leigh wrote, but would still run the risk :)), this is a combination of both roles. Why? I'd attempt to answer that question using the prompts that were given on the course's main page.
When does the act of teaching compromise the role of a facilitator of an online community?
Each time a teacher starts a new online course, (s)he assumes more the role of a facilitator than that of a teacher. It is the first weeks, not to say the first hours that are very often decisive about the course's success. Thus, it seems that right there at the beginning a teacher should try to incorporate facilitator's strategies to comfort each participant in a new learning environment within a new group of people.
When does the act of moderating online discussion compromise the role of a facilitator of an online community?
Generally, moderating a discussion one takes care of its content, relevance and even quality. I believe that a moderator is someone who by his/her 'care' unites the participants, and it is that unifying character of moderator's tasks that makes it possible to compare him/her with a facilitator.
When does the act of facilitation compromise the role of a teacher or moderator in an online community?
The answer to that question would surely summarize my answers above. A facilitator has to make sure that online community members share interests, common grounds and have enough opportunities for interaction - similar to a teacher in an online course (and not only). Simultaneously, a facilitator is 'uniting' participants in a shared cause, problem, issue, topic, etc. This is the characteristic, which I could compare to a moderator.
When are these three roles appropriate in an online community?
The fastest way to answer - whenever an online community emerges. What all these roles share is authority. Whether it is a teacher of an online course, a moderator of an online discussion or a facilitator of an online community, they all have exercise authority over the rest of the participants. They bring in a sense of leadership and care.
I'd try to investigate the skills later on in a comment to this post.