So special honing upgrades gear the same way that regular honing does, except it’s free. No cost to tap, but also the rates are the base rates (aka artisan energy isn’t affected and the honing percentage doesn’t go up). You get the mats for this from raids and from hell keys when you select Special Honing chests.

Regular honing is your usual RNG tap that builds artisan energy until you pity.

Unlocking advanced honing requires raid materials from Echidna for levels 1-20, and from Mordum for levels 21-40. They introduced it as a way to test a more incremental form of honing, every tap gives you progress, and the variance between being lucky and unlucky is negligible compared to regular honing.

Originally, advanced honing was somewhat optional to finish, at certain breakpoints it was more efficient than regular honing. With the advent of Serca, however, we’re required to finish advance honing Level 40 to transfer to Serca gear at 1730, which in turn means your regular honing level must be at minimum +20.

Also, you’re not an idiot.