Since we launched the closed beta, we've also made several improvements to the experience including the ability to set Cortana as the destination for the home button press. Now access to your personal digital assistant is just a button-press away.
You will also see a few differences between what Cortana can do today in her "native" environment (Windows) and in the app on Android. For example toggling settings, opening apps or invoking Cortana hands-free by saying "Hey Cortana" are only available on Windows for now. As with all betas, we are continually improving the experience and will incorporate feedback along the way.
New users will have to go to this link on the Google Play Store and agree to the terms of the testing program before they can download the Cortana public beta.