Why doesn't the keyboard screenshot button work on Ubuntu with GNOME shell?

The PrtScr/Print key still works – but now (U17.10) has different actions bound to it. Go to

Settings → Devices → Keyboard

and scroll down to Screenshots. It shows

If that's sufficient for you and you can remember these keystrokes, then you are done. Else refer to @pomsky's answer and set up a shortcut to launch gnome-screenshot --interactive instead.

Now once you press PrintScreen, gnome-screenshot captures a screenshot and saves it in ~/Pictures/ without asking the user.

You may set a custom keyboard shortcut to launch gnome-screenshot in interactive mode, i.e. when the following window appears providing the user some choices:

and the options to save or discard the screenshot, chose location to save etc. after capturing.

To set the custom shortcut follow the steps from this answer.