Greek yogurt is currently in vogue, and is enjoying a nebulous halo of healthiness. There are certainly some health benefits, but going by an admittedly limited and informal survey of my health-conscious friends and associates, the average yogurt consumer does not know why Greek yogurt might be healthy (or not). Granted, this is not a major health issue, but I firmly believe that no halo should be allowed to remain nebulous -- so here's a quick overview of why you should or should not replace your basic yogurt with a more Grecian variety.
Pros: Greek yogurt is basically made by straining the extra whey out of yogurt. Aside from being watery, whey contains a lot of lactose, the sugar found in milk. This is good if you are watching your calorie intake, or if you are diabetic. It may also have a benefit for those of us who are borderline lactose intolerant, as it has about 45% less lactose.
When you take the whey out of your yogurt, you effectively concentrate all the non-whey bits of yogurt, which includes protein and probiotic bacteria. Most Americans, as it happens, are not lacking for protein, but this may be a benefit for athletes. There are no RDAs for probiotics, but in general a little but more is better than a little less, so Greek yogurt gets definite points on this.
Cons: As I said above, straining the whey out will concentrate the other components of yogurt, including fat. Fat is an important part of a healthy diet, but yogurt is not the best source for it. It contains comparatively low levels of fat-soluble vitamins and essential fatty acids. It does contain conjugated linoleic acid (CLA), an anti-carcinogenic essential fatty acid found in dairy products, but as far as I am aware there are no studies showing that more CLA is necessarily better, and there are potential adverse affects to high CLA intake, so I would not list this as a definite benefit unless you are extremely limited in your dairy intake.
Straining out the whey also reduces the calcium content of yogurt. Calcium helps build healthy bones and prevents colon cancer, and older individuals may want to avoid Greek yogurt in favor or more calcium-rich yogurts. Generally, even Greek yogurt has enough calcium to provide a good percentage of your RDA, so most people will not consider this an important reason to avoid it.
On a related note, be careful when buying Greek yogurt. 'Greek yogurt' is a trendy name for strained yogurt, but it doesn't actually mean much. There are plenty of unstrained yogurts that have added a thickener to get the creamy texture of 'Greek yogurt', and there is no reason that they shouldn't call themselves Greek yogurt. Other Greek yogurts may add whey protein concentrate, which will add lactose back into the product.
Strained yogurt is produced all over the world, and may be sold under many different names, including yogurt cheese, labneh (or labaneh), and dahi. These may all be slightly different in their sugar, fat, and probiotic culture content. If you are travelling in Greece and buy yogurt, odds are that you are buying 'normal' yogurt, as the Greeks usually only strain yogurt for use in certain recipes.
In addition to all these considerations, some Greek yogurt is made with sheep or goat milk, which contain less lactose to start with, or organic milk, which has other health benefits. These variations are beyond the scope of this node, and should be researched separately. Yogurt, it seems, is not a simple issue.