Some people might disagree, or point out that it's perhaps not quite as black and white as that, but in my opinion, classing the AOH as some sort of 'Green' Orange Order wouldn't be far from the mark.
With, admittedly, limited knowledge on the Apprentice Boys, I would have thought they would be closer to the OO/AOH than they would be to a RFB. Loyalist flute bands on the other hand, whilst there will be some who will say they are nothing alike, I think most sane folk see the similarities.