iOS 10 is not gonna change the entire user experience like iOS 7 did but it's going to bring some interesting changes for end users.
Apple is mostly focusing on opening up Siri, it's companion assistant, via a new SDK. This means that apps will now be able to use it. On top of that, Apple is redesigning the lock screen with 3D touch-enabled shortcuts that lets you access your apps' essential functions quickly.
Apple music will also get a facelift with a better library view and lyrics included directly in the app. You'll also find the ability to get rid of unwanted apps like stocks or weather that you can then re-download from the App Store if you ever need to. There's a ton more with a bran new messaging experience, a new Homekit app, unlimited tabs in Safari etc.
Of particular interest to us, Game Center's app is now completely gone on iOS 10's Beta 1. In Apple's documentation, it's clearly written that this is not a bug and that apps should adjust to this change. This doesn't mean that Game Center is completely gone as it still operates the servers and multiplayer functions but you won't be able to access its app for now.
Game Center has just been introduced in WatchOS so it's unclear at this time if Apple wants to completely get rid of the app on iOS or just offer it up on the App Store optionally. What's sure is that, for now, games will be still be working with Game Center, you just won't see the app anymore.