In general, I prefer to use flat (planar) slices through the entire puzzle (interior mechanism included) whenever possible. This means that most of my puzzles end up having a "spider" core, or something similar. Of course, some puzzles with multiple shells may require a more complex mechanism, and possibly a partial or full sphere core to accommodate the multiple levels.
I prefer to design puzzles with flat slices since it gives the puzzles better stability, and a less "squishy" feel. It also minimizes friction since the lowest possible surface area is involved in the turning action. I came to these conclusions after an experimental redesign of my Fracture-10 puzzle. I made a version with a ball core, which allowed me to create space to attach the retaining arms onto the core by screwing them on from the inside. After testing this puzzle though, I was disappointed with the results, and concluded that the original snap together design (which uses flat cuts throughout) is superior. The version with the ball core was harder to turn due to increased friction, and since the parts on the moving slice wanted to first move relative to each other before they would begin to move as a unit (this leads to a "squishy" feel). The spider core version with flat slices has no such issues.
If you consider the Mosaic Cube, it was shipped with a spider core, but was very squishy, and this caused problems in turning. When this was replaced with a ball core, the puzzle was much better.
I think the issue with the Mosaic Cube was not a matter of ball core vs spider core, but of a design that left too much open space on the interior of the puzzle, and allowed parts to be pushed into that space. I don't know if the type of design I prefer could have been used in this case, but if so it might have helped the issue more than the ball core. Even with the ball core fix (which I implemented on my copy), the puzzle is still somewhat squishy, though not nearly to the degree it was before the ball core.
Many of the puzzles I have designed, including just about all in the "Fracture cut" series, use this type of design with flat, planar slices. Some geometries allow the use of screws, while others are better off as snap-together puzzles. Both varieties work very well.
As to cost savings, I haven't seen that one style core is inherently more or less expensive than the other. In my Fracture-10 experiment, the model with the ball core ended costing about $5 less. However, I recently looked back at the original Fracture-10 files, and saw the opportunity for savings there too. After a few changes which will not be detectable once the puzzle is assembled, I retained the original spider core design and was able to reduce the price by almost $10. So that's even less expensive than the ball core design. Here's a link to the new Fracture-10
model for those who are interested.