They don't really serve a purpose. For other games, Origin tracks the achievements and shows them on your profile, but TS4 is not included in that system. They don't give you anything for doing them and your only way of showing them off is to share them on social media to people who probably don't care.
Apart from that, they're most likely meant to provide the player with a sense of fulfillment, satisfy completionists, and provide incentive (if you can call it an incentive) to play the game longer. I think achievements in any game are mostly something that console players tend to care more about.