That part is the same: you're taxed on your worldwide income in EU countries as well - with few exceptions (like Malta).And if you live in Europe, you'll be taxed on your worldwide income. What's the difference?
The problem with US is that you can't go to another (tax preferable) country and claim tax residency there that easily (you can go from California to Florida and pay less taxes as there is no state income tax but you're still in US).
In Europe you could easily move to Cyprus and become a tax resident there and pay less taxes than in Germany for example.