One big difference between the U.S. and most European systems is that in Europe, high school (gymnasium, etc.) is much more rigorous than in the U.S. In the U.S. system, for many students the first two years of college are spent catching up with the things they ought to have learned in high school, but didn't. In substantial part this is because in the U.S. we believe that "everybody ought to have a chance to go to college," whatever their qualifications and background.
In Europe, just entering university is more like at least an associates or community college degree in the U.S.
At the other end, in mathematics at least, in the U.S. you go to 4 years of undergraduate, 4 or 5 years to graduate school graduating with a PhD, grab a one or two year post graduate felllowship and then off you go -- you are a mathematician. As I understand the system in many European countries, after graduate school you get a job at a university as a lecturer, etc., publish a respectable amount of professional level research, work with a famous mentor or two, and only then write a dissertation and receive the degree of doctor of mathematics.
(The U.S. also has that strange pseudo-degree, PHDABD -- PhD All But Dissertation. In other words, you took all the courses and passsed all the exams required for the degree, but did not write a dissertation.) A master's degree from a prestigious European university would be the equivalent.)