I am a very latecomer to the question. Thus, I doubt my contribution will do any good. I try. The Moore Method is one of my favorite "teaching" methods. I have used a variant of it several times and in particular in two courses: Multivariable Calculus and Number Theory. For the former, I designed the course myself (for details see my paper "Moore and Less" : http://www.tandfonline.com/eprint/jGE3QNxcuGzUGj273smp/full), For the latter, I used "Number Theory Through Inquiry" (http://www.maa.org/ebooks/textbooks/NTI.html). Generally speaking, the advantages have been mentioned more or less in the previous answers/comments. Thus, I focus on potential disadvantages.

Playfulness: The Moore Method strictly used is not that much playful. The point is you have a setting in which the materials have been prearranged. This forces you and your students to stay on a predesigned track, and as a result, hinders useful and playful jumps. Let me give an example. Suppose you start with some examples of primitive Pythagorean triples. The most famous ones are (3, 4, 5) and (5, 12, 13). Observation: the difference between two of the numbers is one. Having characterized Pythagorean triples, it would be natural to move to Pell equation. However, since the materials have been prearranged, you should continue with Lemmas ., Theorems . , all directly related to Pythagorean triples.

Naturalness: Again, this is somehow related to the prearrangement. As a designer, you feel that you should provide some backgrounds to help students to prove a certain theorem. What you provide is natural for you since you already know the proof of the theorem. However, it is not so natural for most of the learners. Let me go with the previous example. Moving towards the theorem characterizing Pythagorean triples you write (your students read): “It turns out that there is a method for generating infinitely many Pythagorean triples in an easy way. It comes from looking at some simple algebra from high school. Remember that … “. It gives me a very bad feeling to behave with my students in this way, to say the least. The problem arises even for the lecturer when others have designed the course. An arrangement that is natural for somebody else is not necessarily natural for you.

Forcefulness: Your students are forced into forward thinking. There is a theorem (again take the previous one as an example). You are somehow sure that your students need some help to prove it. Where do you provide that help? As a Lemma before the theorem!

Connectedness: This one is very strange and paradoxical. While you are connected with individuals and/or small group of students working together, you lose your connection with the class as a whole.

There are some other points. I stop here since my answer is already too long. Moreover, I couldn’t find suitable words ended with “ness” to describe the other points ☺