Believe it or not "Vassilis" is the inofficial form of "Vassilios". Not much of a difference one would say, but in fact the difference is so big to Greek ears that I do not know anyone with this name who would introduce themselves (at least in a Greek audience) using the official version.
In case you are interested. There are a few issues with Greek names which complicate things. The most evident is the spelling of a Greek name using Latin letters. Besides the obvious rules such as turning "α" into "a" and "μ" into "m", in most of the cases there is room for different ways of assigning a Latin letter to a Greek one. For example some people would turn "κ" into "c" while others would turn it into "k". Similarly "η" can be turned into "e" or into "i". (The latter possibility makes sense because unlike the English pronunciation "η" in modern Greek sounds as "i" in "bit". For this reason more and more people decide to turn "η" into "i". Personally I am in favor of "e" though.)
It seems as a good idea to establish once and for all a single way of assigning a Latin letter (or letters) to a Greek letter. In fact such an official rule is being applied in Greece during the last decades to official documents such as passports. So problem solved, right? Wrong! While this rule keeps bureaucrats happy it creates more problems than it solves. The most evident issue is what happens in case of a Greek silent vowel. For example "ει" in modern Greek is pronounced exactly as "ι". The official rule will turn "ει" into "ei" and not into "i" creating thus a totaly different pronunciation. Similar problems arise with Greek consonants. The easiest example is the letter "σ" which is turned into "s" and never into "ss" resulting in many cases to the unwelcome sound "zeta". Pronunciation is not the only issue though. In fact sometimes the official rule is very good in preserving the phonetics, but this is usually on the expense of what is historically accustomed. For example the typical English pronunciation of "Sophocles" is not 100% accurate with respect to modern Greek. But let's be honest: "Oedipus the King, by Sofoklis" would look strange to a lot of people.
To see how ridiculous the thing might become consider the fictional example of a Mr. John Michael who for some reason is applying for a Greek passport. His name would first be spelled in Greek preserving the phonetics as much as possible, and then it would be turned back into Latin spelling using the official rule. You can be sure that Mr. Tzon Maikol (or Maikl depending on his accent) wouldn't appreciate the outcome. Fortunately some people in the Greek administration are smart enough to allow some rare exceptions from the official rule in order to avoid such ridiculous circumstances. The cases however which produce a peculiar (under one's perspective) Latin spelling are so many, that in my opinion it doesn't make sense to try to impose any official rule at all. For more information on transliteration and transcription see here.
In case you are really interested. Unlike English names, the Greek ones are subject to grammatical cases. In practice this means that the ending of a Greek name changes depending on the way it is used in the sentence. For example the correct way to address me is "Vassili". The version with the "s" at the end is just the nominative case. (Perhaps the main reason that "Vassilis" sounds to Greeks so far away from "Vassilios" is because they differ not just in the nominative case but in fact in every grammatical case.) And yes, your good friends "Giannis", "Dimitris" and "Andreas" should be addressed "Gianni", "Dimitri" and "Andrea". Although every Greek addresses me the correct way I am not expecting that everyone that I meet is aware of Greek grammar. So calling me "Vassilis" is fine with me. Some people do mind though. I know cases where people introduce themselves using the vocative case (which is grammatically wrong) just to make sure that they will be addressed the correct way.back