Mozilla Persona is private by design, and users cannot accidentally overshare information. The website owner can only access as much information as the user chooses to pro-actively share. Unless the website has a membership listing or social features, no one will know you are using that website. You are also able to create new identities for websites as quickly as you can create an email.
Google+ Sign-In defaults to a more social experience. With the user’s permission, gives the website owner access to profile data, a user’s circles, and possibly the ability to make API requests on the user’s behalf.
This is one area where the scope of Google+ APIs is a blessing and a curse. One on hand, you can create a rich and social experience, but there are lots several APIs to learn, including some that might do the same thing in a slightly different way. However, everything is housed under the same roof. Want to use the Hangouts API with your app? No problem. YouTube API? Sure.
That social data does come with a cost. The Google developer guidelines state that you must give the user a way to disassociate the account from your application. If a user chooses to delete his or her account with you, you must delete any information you retrieved from Google’s APIs. Even if the user chooses to maintain an account with you, you must update the account information you cache in a timely manner. For example, if a user changes his or her profile picture or removes some data from the account, you need to reflect those changes in your datastore.