I tried out both with push off. It worked on each one like it should so you have no worries there.
The only difference that I see on Find My Friends is that your can "follow" more than one person. It still requires entering the password - however it's your password not the other persons. However, I think if you have a passcode set on your phone that entering your password into the app isn't required. In FMF, you can also "hide" when you'd like and if anyone is using it for their kids, restrictions can be placed where they can't turn off the FMP feature without a passcode. FMF has other features like being able to designate an address as home, work, gym, etc and it also shows their contact info that you have store in your phone. It does require an invitation to be accepted. An email is sent as well as an invite within the app.