How is the training for MS students different from MBA students? How do universities train MBA students differently from MS students? Statistics and quantitative reasoning are vital parts of the MBA as analytics have become crucial to many aspects of business. In what sense does an MS in Statistics have better training in analytics than an MBA?
How do MBAs add value to a person's career? Is their broad (if not deep) command of statistics, economics, finance, etc., the reason behind their astronomical salaries? Is there specific training that makes them 'managerial'? If yes, what is that training?
An important aspect of MBA training that I haven't seen mentioned in other answers is networking.  A huge amount of getting things done is business is about networking, and while you will not find a formal course on networking in an MBA program, most good MBA programs will have a lot of networking events and opportunities connected with them.  MBA students are also often grouped together in cohorts or otherwise treated much more as a unified group of students than MS students.  All of this adds up to a very important extra-curricular curriculum component that is often a key differentiator for success or failure after business school, and is largely neglected in the more technically focused track of MS students.