There are certainly achievements in the game. It's similar to how Achievement Diaries were known before or how they are in Old School RuneScape, but with some changes. Check out the Wiki article about them here:
I wish there was a way to know you're in the good old days before you've actually left them. (S09 EP23)
As posted above, the achievement system still exists and has been expanded, with specific rewards now available for areas like the Elf Lands and Wilderness. Spearmint30's link should help explain it further.