In public free roam, you share the world with a number of other players.
In private free roam it is just you, or your posse if you made one, having their own instance of the world.
You have quest-like goals you can accomplish in free-roam:
Clearing gang hideouts, collecting, hunting, etc.
Or you can enter different types of multiplayer games:
Deathmatch, capture the bag, etc.
You can meet up with other players in the world and add them to your group and do the goals together or enter the games together.
Or you can go around and do it alone. Although you have to watch your back because there is nothing stopping pking and spawn camping. So being in a posse is safer.
If you pick up weapons in free roam, you keep them until you log out and end your session. All unlocked weapons you keep indefinitely. You unlock weapons, character skins and mounts by doing matches or completing goals or just running around killing players and NPCs. They all give you xp.