I have seen a few Warlock Bonds (Hunter Cloaks/Titan Badges) but their only difference appears to be their appearance and a cryptic parable written beneath them. I know that the class items from the factions have something to do with converting all Vanguard and Crucible advancement progress into progress towards advancement in those factions, but what about the class armor from the Speaker or Vanguard shop? Do they affect rep gain, drop rates, or otherwise change anything? Does a rare Bond from the Cryptarch offer something different from an uncommon Bond (especially if there are no features listed)?
If the class armor is not from a faction (vanguards don't count here) then it's just for decoration. If it is a faction's armor bit then you'll earn rep for them instead of for vanguard or generic crucible faction.
In either case, rarity has no particular effect.
Note that even with faction armor equipped that turns vanguard rep into faction rep, you'll still get vanguard marks for missions that give them. Since faction vendors only take crucible marks, this may affect your strategy when gaining rep.