Moai is opensource, lower-level engine that appeals more to those that want full control and customization. Tooling is mostly "bring your own editor/ide" (Intellij has a good Lua plugin[1] and there's also Zen Braine studio[2]). Personally, I would rather use it over Corona, but those aspects are appealing to me, but could be off-putting to others. You'll be looking through the Doxygen documentation and perhaps even the C++ source at times to figure things out in Moai. The plus side is much of the design/best practices of Corona Game Examples (or any other Lua Game framework, like LÖVE) can be used to model your Moai development, omitting the API differences.
Corona is more for those that want a high level game engine and all the tools for it built in as well as more beginner friendly (larger community, documentation, ease of use). The drawbacks are limitations from the source being closed and in the past there were performance issues for Android/iOS, but I have not tried Corona lately so it may have improved in that area.
[1] https://bitbucket.org/sylvanaar2/lua-for-idea/wiki/Home
[2] http://studio.zerobrane.com/