What is the purpose of this 4-pin interface on SATA HDDs and why doesn't it exist on SSDs?

Those are jumper settings, similar to the IDE drives you mention, but for SATA specific options depending on the drive maker. For example, on this Western Digital support page for mobile drives, it shows two options:

• Reduced Power Spinup - RPS: Used by external hard disks to reduce spinup current and allows it to work over USB interface.
• Spread Spectrum Clocking - SSC: Used when your SATA controller doesn't support SSC by default.

Also, it doesn't exists on SSDs because there is no spinup or spread spectrum related to solid-state devices.

@Adrian Cox mentions in the comments below, a different pin function for Seagate drives.

According to what i can find its a Manufacture specific firmware terminal that aids in HDD recovery. PC USB Terminal adapter There's a few images on this particular page that shows how to connect a USB cable to those ports using Manufacture specific adapters.