There are differences between logging out of and exiting the Fongo Mobile app.
Android
Exiting the App
Tap Account, then tap Exit to simply exit the application without logging out. This will enable to you to still receive push notifications about messages and incoming calls, but the application will no longer run in the background.
Logging Out of the App
Tap Account then Logout to sign out of your account. This will allow you to sign in with a different account and you will no longer receive any push notifications for the previously logged-in account.
iOS
Logging Out of the App
Double-click the device's Home button or swipe up from the bottom of the screen to see the list of running apps, and swipe up on the Fongo app to close it. This will enable to you to still receive push notifications about messages and incoming calls, but the application will no longer run in the background.
Exiting the App
Tap Account, then tap Logout. This will allow you to sign in with a different account and you will no longer receive any push notifications for the previously logged-in account.