1. You don't have to take the GRE. Instead, applications are heavily made up of your undergraduate research, which European schools place a lot more emphasis on. This is something I really give European universities a lot of credit for: taking the time to closely examine the research and work you've already done as an undergraduate. In sharp contrast, American schools tend to focus on your resume/CV, transcript, and GRE scores. With the exception of personal statements, applicants to American colleges aren't really given a chance to showcase their material academic achievements and illustrate where their passions lie. The European applications seem more concerned with the scholar as individual, than with standardized assessments such as GRE scores.
2. European universities have been around a lot longer. Uppsala is the oldest university in northern Europe - founded in 1477. Cambridge was founded in 1209, and Oxford is so old that it's year of inception is unknown - definitely 1096 or earlier, making it the second-oldest surviving university in the world. All over Europe, universities are preserved in beautiful buildings which have been standing for hundreds of years. The presence of so much history, so many previous scholars and students... these are academic institutions which have withstood and born witness to countless eras!
3. MA programs are only one year. This isn't the case with all of them, but a lot of MA programs in Europe are only one year (check out the list of King's College MAs for loads of examples), whereas in the States the typical length is 2. The degrees are the same, but the time (and money!) you put in are less. Some universities even allow you to choose between a 1- or 2-year version of each masters program.
4. Tuition is cheaper, or even free. Grad school is so much cheaper in Europe (you just have to figure in living expenses, of course). In some countries (particularly Scandinavia) education is heavily or completely subsidized by the government, which means tuition is unbelievably low by US standards. In Norway, which is not part of the EU, the government COMPLETELY subsidizes education. This means that if you wanted to get your MA at the University of Oslo, even as an international graduate student, it won't cost you anything (except living expenses in Norway).
5. It opens up job opportunities abroad. Going to university in Europe gives you a fantastic leg up if you ever plan on living and/or working in Europe. Often after getting a degree in any foreign country there are additional perks, such as the ability to get a work visa more easily.
6. You interact with a truly "international" student body. In the European Union there's so much international connectivity and the universities really embrace this. Many European universities (Uppsala is no exception) have very large international student percentages to their student bodies, facilitating that constant cross-cultural discourse that is vital in today's world. So if you go to grad school in Italy, you won't just be hanging out with Italians. You won't even just be hanging out with Europeans! European universities are fantastic global microcosms, and your peers will truly come from all over the world.
7. Programs are often offered in English. Because of this emphasis on creating a multinational student body, universities in countries with languages that are not widely spoken (such as all the Nordic countries) offer programs in English. Of course, it's always a good idea to try to learn as much of the local language as possible when living in a foreign country, and building language skills is a vital CV-builder in today's world.
