The friend battle system uses Game Center as the contact list. If you add another player as a friend in Game Center and then refresh your HI app, their name will appear to invite them to a PVP battle. From there PVP functions the same as it does in matchmaking.
The reward is also greater for a friend battle once every 24 hours, awarding 777 diamonds for a win vs. the 400 max in matchmaking. However the award is only available once per person per 24 hour period.
Hope this helps explain