The only advantage I see is that it actually seems to keep a better handle on the status of the process/service. The old-style unit scripts would often get out of sync and not realize that a process had died, or if they did they would repeatedly respawn a service that would just die again. Maybe that was less of a problem in later years than I experienced earlier, but it was there.

The whole init.d system felt very ad-hoc with every script working a little bit differently, giving different output styles, etc.