Firstly, speaking in such general terms about the USA and the EU isn't going to be very productive; both regions are massive and diverse, and the lifestyle, quality of education, and cost of living are going to vary tremendously within each.
You need to do your research and consider programs individually, although I can make a few generalizations about the US and EU that might inform your decision.
Compared to Europe, America has horrible public transportation and is extremely car dependent; public transportation exists, but it is unreliable and often times dangerous when compared to Europe - this is because most people don't use it. This can be very limiting for poor students who can't afford cars to get around. European cities generally have great public transportation and you wouldn't be as limited in your ability to travel in the absence of a car.
Also, it is true that Masters programs are the norm in the USA - but they aren't a requirement. But PhD programs are very competitive here - I don't know if they are or not in Europe/Germany.
If I were you, I would do my research on individual programs that interest me in both the USA and the EU, and apply to the best ones that I thought I had a chance of attending. Then, I would pick the program with the best financial aid offer, considering the program's reputation and the region's lifestyle as important secondary factors.