Fage is one of the few brands that doesn't add carbs to the 0 fat product (or maybe it's one gram of carb more per serving). Since the main reason to avoid fat free is the weird crap they usually put in to compensate for the missing fat, Fage is one of the few brands where the zero fat is okay imo. But if you have access to both, might as well eat the full fat one.
"Right is right, even if no one is doing it; wrong is wrong, even if everyone is doing it." - St. Augustine
Who says back fat is a bad thing? Maybe on a hairy guy at the beach, but not on a crab.
I usually get the 0% or 2% Fage greek yogurt--not because I'm concerned about the fat, but because the taste of the full fat variety is too rich for me.
I don't understand the comment above about the use of artificial sweetener (particularly in plain yogurt). I don't believe that's the case, at least as far as Fage's products go...
Grab some of the Yoplait, Dannon, or other popular brands and do a side-by-side. Low-fat/light/fat-free/etc. = increased sugar or artificial sweeteners in most cases. The big NYT article that was linked to all over the forums about General Mills, Coke, and other big food names specifically call out that the fat-free Yoplait yogurts have more sugar in it per serving than Lucky Charms cereal.
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If you want fat free yogurt go with Icelandic Skyr. They really have the taste and texture down. It's a lot thicker than the nonfat greek yogurt and was designed to be made from skim milk originally anyway.