I see both version and was wondering what the difference between the two is?
These are two different cheeses. I don't know how they came to be called by the same name, but they have a very different taste and texture.
The Greek-style Feta is a subtype of the standard cheese on the Balkan peninsula. When I say "standard", I mean that it is so entrenched, there is not even a word for it - linguistically, "cheese" just means this type, and all other kinds are an exotic afterthought.
This Balkan-type cheese is a semi-hard white cheese, for which the curds are ripened in a salty brine, without culturing. It is crumbly, and has a characteristic aroma, especially when well-ripened and when made from sheep or buffalo milk. I am not sure whether the Feta PDO only covers the sheep-milk type, but the process is suitable for all milks. It is quite salty, and melts only partially when heated.
The Danish Feta is an entirely different beast. It is a soft cream cheese, similar in texture to American cream cheese. The taste is a bit stronger than American cream cheese. Typically, it is not marinated, even though the product you show is sold in marinade - but this is more of an "additional treatment", not part of the cheesemaking process. The only similarity to Greek Feta is that is also white in color, and that it happens to bear the same name (at least until the PDO laws and their application get sorted out).