What are the differences between the Tower factions?

Solution 1:

The stats of the items for each faction has differences.

One strategy is to figure out which faction has the piece of equipment you most want and work on that faction. For example only Future War Cult and Vanguard have Hand Cannons, so if you want one of those you can decided between them.

You should also be aware that while wearing their colors you'll turn Vanguard and Crucible faction reputation rewards into the faction rewards you chose. You will not turn Crucible or Vanguard marks (the currency) into faction marks.

You can't turn Vanguard Marks into Crucible Marks. Faction items (except Vanguard) can only be purchased with Crucible marks.

Solution 2:

as far as I can tell it has to do with stats, Intellect, Strength, and Discipline each one has armor that gives different buffs FWC gives int and dis for example