From 1834 to 1976 there was an attempt to impose "Katharevousa" ("purified" language -- an attempt to "correct" centuries of natural linguistic changes) as the only acceptable form of Greek in Greece. After 1976, "Demotiki" (speech of the people -- the modern Greek) was finally accepted by the Greek government as both the de facto and de jure forms of the language.
Unfortunately Greek names were not included in this change--possibly to avoid bureaucratic confusion. As a result there is a different spelling and pronunciation of names in formal documents and in everyday life. For example my name, as far as the government is concerned, is "Vassilios", while everyone is using the modern form "Vassilis".
If that was not enough, the Greek Foreign Office decided to translate Greek names in English in a consistent manner: letter-by-letter. As a result, the name written on my Greek passport is "Vasileios".
Most people don't know that, even with modern Greek there are four variations of each noun and name, depending on their position in the sentence: nominative, accusative, genitive and vocative. Nominative is the subject in a phrase, accusative is the object of the verb, genitive is the possessive case, and vocative is used, for example, to call somebody. So, the nominative case of my name is "Vassilis" and all other cases are "Vassili". (However this doesn't generalize to all Greek names)