Earth's moon goes through its complete cycle of phases about 13 times per year. This is, of course, where we get the idea & term for 'month'.
Something that I haven't really noticed anyone making note of when discussing a moon in a story, or shown in a movie/TV show is an exoplanet's moon rotating and showing different 'faces'. Our moon doesn't do that due to how it was created - being ejected after another massive object colliding with Earth - so its axial rotation period matches its orbital period. But that will not be true with every planet's moon(s).
So even if the planet has a permanent full moon (however it is done) it could have different 'faces' as it rotates on its axis.
Using my idea above, imagine that Jupiter currently was in an orbit (around the sun) at only 120 million miles from the sun. We would be able to see the Great Red Spot traverse across it every 9 hrs, 55 min, 30 sec. In earlier times people might have thought of it as God's eye turning towards us. There is interesting potential in such an arrangement.
If it is a moon the terrain could have features which people could think of as all sorts of things. We used to refer to the "man in the moon" due to the lunar features vaguely resembling a man's face. With a different moon, it opens up the author's options for what cultural influence the moon's features could have.