They don't drop anything or progress the evil biome boss's summon.

My (wholly unprofessional) theory is that there's some trigger that makes the natural/shrine orbs switch usability, having to do with which group of orbs is broken first.
In my most recent world, I broke the two in the ruby ring shrine, and then the natural ones became unusable. In a previous world, I did summon the evil boss without issue, but when I found the shrine a while later, the orbs inside did nothing.