This is what I've discovered thus far regarding factions and their gear. In order to get any reputation for a faction, then you need to equip their aesthetic item. These can be bought from your chosen faction merchant or even be found. When you have this item equipped, any reputation that you would receive is transferred into faction reputation. Any Crucible or Vanguard reputation from bounties are going to go toward your faction now. Each faction has their own stat boosts (Intellect, Discipline, and Strength). Intellect lowers the cooldown of your Super. Discipline lowers the cooldown of your grenades. Strength lowers the cooldown of your melee. Dead Orbit items focus on Discipline and Strength. This allows players that use their grenades and melee attacks often to thrive. Future War Cult items focus on Intellect and Discipline. This allows players that like a quick Super and use grenades often to thrive. New Monarchy items focus on Intellect and Strength. This allows players that like a quick Super and use melee often to thrive. Other merchants that sell armors have different stat boosts on their items. Factions are the only items that you're guaranteed to have the same stats to help you.