Like most others have stated, it really depends on your budget and your likes. Given that this is part of an around-the-world trip, I'm assuming that budget is not as much of a concern.
If you like scenery, it's hard to say if Moorea or Bora Bora is nicer. The water in/around Bora Bora is unsurpassed, but the combination of jagged mountains, deep bays, and water in Moorea is to me the best overall combination.
Re: activities, if you like water both are good, but Bora Bora to me takes the prize. If you like scuba/snorkeling, Bora Bora has great places within and outside of the reef. Moorea is great too, but Bora Bora wins (to me).
There are more high-end lodgings on Bora Bora than Moorea(although there are a few good budget places too), though Moorea has a few nice high-end places as well.
Practically, Moorea is much easier, cheaper, and quicker to get to than Bora Bora (plan on losing about a full day getting to/back from Bora Bora), though the trip to Bora Bora is pretty nice in and of itself - the view from the flight and the mini-tour provided by the airport water taxi are great.
If I had only 6 days, it's VERY close but I'd have to go back to Bora Bora again. Have Fun!