Is getting into MSc programs in Europe easier than PhD in US? PhD programs in US usually include an MSc degree and many of the admitted students don't have an MSc. In Europe however, with just a few exceptions, everywhere requires an MSc for a PhD. There are also quite a few differences in the structure of the undergrad in US and Europe.

I was wondering if it is easier for someone with a US undergrad degree to get into an MSc in Europe than PhD in US? Are the PhD admissions in US more competitive? 

If I have an offer from a couple of top US schools (by "top" I don't mean Harvard and MIT, I mean around a dozen (national) ranks below that... say, top 30-40 in global rankings), should I be confident that I can get into one of the few MSc programs in Europe I've applied to? (The programs are top/best programs in Europe, but the universities are essentially on par with the US school in global rankings. Perhaps some of them have a better global reputation in the specific field.)

Note:

1 - By Europe I am referring to Western Europe, excluding UK. More specifically, Germany, Switzerland, France, Belgium, Holland, Austria.

2 - The field is Mathematics, if that makes a difference. (I think it might, given the reputation of places like Heidelberg, Bonn (Hausdorff Institute), and the universities in Paris have in Mathematics.)
Master's education in Europe is largely considered an "undergraduate" degree that follows the bachelor's degree, rather than being the prerequisite for the PhD. Thus, admission to a European master's program tends to be much less competitive than PhD admissions in the US. Now, if you're applying to a top program in Europe, it can be as difficult and competitive as in the US. But the average master's program in Europe has a lower bar to admission, so long as you've done the necessary undergraduate studies considered a prerequisite of the bachelor's degree program.

The rules about what is considered acceptable varies, but a few rules of thumb apply:


In a “consecutive” master’s, which requires a bachelor’s in the same discipline, you need to be able to catch up with coursework in less than one year (sometimes one semester, if the program is competitive). 
In a non-consecutive master’s, you just need to have a suitable bachelor’s degree to enroll.