What a wonderful question!
Game Center for the longest time was a standalone app that players could reference to see achievements, friends, etc...With the release of iOS 10 the functionality changed greatly. This functions much more like as service now instead of a standalone product. Game Center stats and services are now held within each individual game that supports it. Here is a little further explanation below:
Game Center is Now a Service
So what this means for you is that while the app is gone, its functionality is still there. Think of Game Center as a service rather than an app. So for all gamers out there, your Game Center Service continues to track your best score(s) on a leaderboard, compare your achievements, invite your friends to play, or even start a multiplayer game through auto-matching. And you still receive notifications from Game Center, provided Notifications for Games is set to ON (Settings>Notifications>Games.)
Now everything that the Game Center app did is available directly in your game’s individual app. Check leaderboards, your achievements, view challenges, send challenges, and see your turns–all inside the game app itself. If you want to manage your Game Center account, you do that in Settings.
You can still reach part of the Game Centers abilities by using the settings application and scrolling till you see "Game Center." Basically, you can view your profile, remove friends, and have the ability turned on to invite players around you for multiplayer games over Wi-Fi & Bluetooth. The functionality is much more limited.
The points are more for bragging rights against your friends and unlocking fun achievements for certain games!
Let me know if you have more questions and i'll be happy to address them!