They have a very good point about comparison strands being anybody's guess. Trust your own judgement by learning more about Akoya cultured pearls and take the time to check out freshwater cultured pearls while you're at it. There are a lot of Akoya strands out there that approach or equal top Mikimoto quality. (Search Hanadama)
Using the search feature of this website will bring up a lot of reputable online pearl dealers. They have excellent photos if you want to see what else is being offered. If you want to see lesser quality, or see what we mean by AAAA being meaningless, just do an eBay search.
I can't speak specifically to Mikimoto treatment. The usual treatment: Almost all Japanese Akoyas are bleached -- even the pearls that will be dyed can be bleached. They are tumble polished and can also be buffed with or without polishing compounds. This can remove some of the nacre, which is already fairly thin. Most Akoyas are "pinked" -- given a bath in dye. This enhances color and may contribute to a rosy overtone if the pearl color is already on the pink side.
At this point, only the producers know what kind of treatments, if any, are added. Mikimotos look that way for a reason. It could be something special in the tumbling and buffing, or it could be some kind of secret treatment to enhance the lustre.
Japanese Akoya pearl nacre is fairly thin and will wear off eventually with daily wear. An alternative would be to get Chinese Akoyas, which tend to have a much heavier coating of nacre. Or, get the finest freshwater pearls, which are entirely nacre, not just nacre coating over a bead. One thing is certain -- Mikimoto pearls are very good pearls. But they are costly and there are a lot of fine alternatives.
You were only asking about what makes Mikimoto different, but somehow I find myself up on my soapbox as usual. I
am a pearl cheerleader.
The journey to find the ultimate pearl necklace can be the best trip of your life.

Warning -- pearls can be addiciting!
Good Luck,
Blaire