Production wise the difference between a quality puzzle and the rest is huge. Junk puzzles are inaccurate, and frequently glued up in the solved state. That means when you take them apart and try to put them back together, the fit between the pieces will reveal the solution…the wrong placement or orientation will not fit well. Shoddy worksmanship literally ruins the puzzle.
The only way to make a great puzzle is to maintain absolute accuracy in all angles and dimensions. This is very tricky when the wood moves every time you cut it. Quality puzzles tend to be largely milled from solid pieces of wood for strength and beauty…cheap puzzles will have more smaller pieces glued together which introduces the potential for failure over time. Finally, quality puzzles will use traditional woodworking joinery to make long lasting joints between pieces. Wood never stops moving with humidity and temperature, and over time will stress and ultimately break inferior joinery. The most common thing I see with cheap wood puzzles are sticks glued end grain to side grain. This is a poor joint and will likely fail since it was almost certainly not clamped when glued. A quality puzzle will use joinery such as shoulders and splines to reinforce the joint and create heirloom quality work.
Design wise, good puzzles take advantage of assumptions the solver is making. Great puzzles will create those assumptions and then use them against the solver. Great puzzles can also do something unexpected or tricky…a really great puzzle can even be witty or humorous in solution. They have a "snap" to them. Perhaps it's an intuitive leap, or taking advantage of an assumption but the great ones give that "ah-ha" feeling.
For more thoughts on what makes a good puzzle, check out this excellent article on the Metagrobologist.