Achievements are neat as an ingame concept. They can, as mentioned, drive players to play the game in different ways, be used to unlock bonus content, track completion, etc.
Achievements as a platform system though, are questionable. Making achievements into a generalized system invites the line of thinking where "why doesn't this game have achievements" seems like valid criticism. Which leads to developers throwing in achievements just to tick that checkbox, with no thought or purpose behind it, invalidating the good things that achievements bring to games, and bringing forward the bad ones - completionist achievement-hunting, obsession, boasting, etc.
It's a bit like the difference between faith and religion. The latter is supposed to promote the former, but instead just leads a lot of people into following the motions without actually understanding what it means and why it's important.