Pinkberry was the first to land in NYC, with Red Mango quickly following suit (although it appears Red Mango was actually around before Pinkberry, but the latter made it to the US first).
I tasted both the original and green tea flavors at each shop (Pinkberry also has a coffee flavor I didn't sample).
Pinkberry's plain flavor (called "pinkberry") is tart, very similar in taste to plain yogurt with just a bit of sugar. I preferred their green tea; the macha, a delicate flavor, mellowed the yogurt's tang a bit. It paired nicely with the coconut.
The texture on both flavors was icy; not in a granular way, but it's very clear that there's no fat to add a silky mouthfeel.
Red Mango's original with blueberries.
I really liked Pinkberry -- different, light, refreshing -- until I tasted Red Mango's yogurt. Their original is tangy, but not as sharp as Pinkberry's, slightly sweeter and has a light vanilla flavor; it's a winner. Their green tea packs a bigger macha wallop, too -- while Pinkberry's green tea was very light, Red Mango's macha is much more present . Additionally, both Red Mango flavors had a creamy, silky texture; it's hard to believe there's no fat in it. If you look at the pictures, you can even see that the Red Mango yogurt is more lush than Pinkberry's.
The verdict? Red Mango, all the way. (It's cheaper than Pinkberry, too.)