I have been to both. They are quite good, and strong in some particular areas:
BSM: combinatorics, number theory. They also have good courses in measure theory (the book we used is big Rudin chapter I+II), intro to topology (the first half of the course is in point set topology, and second half is in fundamental group), functional analysis.
Math in Moscow: algebra, number theory, topology, knot theory, algebraic geometry. You can easily have independent studies in any subject. In my opinion, the courses in Moscow are harder than in the BSM program. People live together in 1 floor, cook in the same kitchen (even go to the same toilet
) ), so for me, there was more fun.
I don't know how hard to get in, but I suppose it's quite easy. People I met varied a lot in their background. There were first courses in algebra/analysis in both programs, and there is a number of people taking those classes. There's another study abroad program in Germany, tuition is cheap, but the cost of living is high.
There's a scholarship for the Math in Moscow from the AMS. (for students studying in the United States), and also a scholarship for students studying in Canada.