It's typically caused by moisture getting into the brick either from the inside or outside. Here's a link to a good article on it: http://www.masonryinstitute.org/pdf/612.pdf

Thanks very much for that article. It sounds like with this condition present, we probably don't want the chimney sealed because then the crystals are trapped by the sealer and can damage the brick?

I assume it is more likely that water is getting into the flue through cracks in the mortar rather than through the brick? Or is some brick actually pretty good at absorbing water?

If you didn't have leaks, I would say that the efflouresence is primarily being caused by the bricks absorbing some of the rain water. But with the leaks being present I think it's more likely that the water that is getting in through cracks in the mortar (although from the picture the mortar looks pretty sound) is sitting on the tops of the bricks and sinking in. Are you sure that the water isn't getting in from the top of the chimney, or the ledge shown in the picture, and running down the inside? Sealing the bricks if there is still moisture in them is not a good idea. You might want to get another mason out to give you a second opinion.