From my limited understanding of yogurts the benefits of Greek yogurt over your standard yogurt are;
#1 Greek yogurt is much higher in protein, as much as twice the protein content as brands such as Dannon.
#2 Greek yogurt is conciderably lower in carbohydrates by as much as 50%
#3 Greek yogurt is thicker and creamier due to it being triple strained as opposed to standard yogurts which are typically strained twice and often time have thickening agents added, whereas Greek yogurt requires no thickeners making it a more substantial and appealing yogurt.
#4 Greek yogurt has much less sodium, as much as 50% but maintains a full-bodied taste.
#5 Easier to digest because is contains less lactose.
#6 Versatility, Greek yogurt can be used in cooking with much more versatility then standard yogurt.

http://www.youtube.com/watch?v=jmJ65_q2jfA

A disadvantage to Greek yogurt is due to the triple straining some of the calcium is removed.
Standard yogurts have two to three times the calcium as Greek yogurt.
Also if your not making your own yogurts then store bought Greek yogurt is conciderably more expensive, so I reccommend making your own, its really not that difficult once you learn the basics.

http://www.youtube.com/watch?v=0jzXUpnz5kc&feature=fvw

From what I can tell it tends to be more creamy than regular varieties of yogurt. It also seems that it can come from either sheep's or cow's milk where regular varieties tend to stick with cow's milk. The second link has more details on the specifics of the process if you're interested.