To be more specific, group dungeons are designed for 4-players teams of a tank, healer, and two damage dealers to defeat the bosses. They have Normal and Veteran variants. Most bosses have between 500k-2 million health on normal difficulty, about the difficulty of a base game world boss. They grant a skill point on completion of the quest. These dungeons can be accessed through dungeon finder or through the world.
Public dungeons are like more involved delves, with larger mobs and several bosses, including one group event. Most bosses have about 200-300k health. They don't require a group with set roles, do not have veteran versions, and are accessible through the world, not dungeon finder.