With iOS 10, Apple made a fundamental change to the way the Touch ID sensor worked, and it's a particularly annoying change.
Previously, you just had to tap the Home button, and then you were into your device. But with iOS 10, Apple tweaked it so that you are now able to unlock the device and still be on the lock screen. You have to tap the Home button again to get to your apps.
Apple made the change so that people no longer blasted past the pile of notifications that piled up on their lock screens since the last time they used their phones. But the change also put a speed bump into their daily workflows, as it forced people to relearn a process that had been ingrained into muscle memory following years of repetition.
It seems that Apple now acknowledges that this was a mistake. In the Public Beta 2 of iOS 10, it has offered users a way to make Touch ID work as it did under iOS 9.
You just need to know where to look for the setting. And, as with a lot of things in iOS, it's well buried.
If you're running iOS 10 Beta 3 or Public Beta 2 and want to make Touch ID work like it did under iOS 9, then the setting you are looking for is called Rest Finger to Open, and this can be found under Settings > General > Accessibility > Home Button.