Is the gap between lower- and higher-ranked graduate programs in math really that big? Note: a question similar to that I'm going to ask was discussed here, but answers to the questions I'm going to ask were not covered there.
Basically, I'd like to understand how significant is the gap between lower- and higher-ranked graduate schools in mathematics. I'll be referring to the USN graduate schools ranking. Let me fix the notation straight away. By 'higher-ranked schools' (which I will also refer to as 'top schools') I will mean top 10 (or if you want top 20) and by 'lower-ranked schools' I will mean schools ranked 20-40 (all according to USN).
What I understood from the question I provided the link to above is that being amidst 'top students' at a higher ranked school is more beneficial than being amidst 'average students' at a lower-ranked school. Also, higher-ranked schools may be more diverse and provide more opportunities in meeting people from other top schools as well as in obtaining a job in the academia. 
But I still have a couple of questions. 
The first one is about the first two years of the study. Is, in general, the instruction level at lower-ranked schools worse than that in higher-ranked ones? Also, are graduate corses at 'top schools' more difficult to master and to pass? If so, does it imply that one needs a better preparation (i.e., a stronger mathematical background) to succeed in a top graduate program? Also, does it imply that the students enrolled in a top program will eventually have a better mathematical background that will make it easier for them to conduct research? If you have anything else to say about the coursework at top universities in comparison to that at lower-ranked universities, I would appreciate it.
Secondly, in the question I referred to above (or elsewhere), some people mentioned that exposure to new ideas in various branches of mathematics (which is one of the advantages of top programs) is a consequence of the size of the department and the 'quality' of faculty/post-docs/students. Whereas I do agree that students at top universities are more knowledgable and creative, I cannot see why the other assertions hold. Correct me if I am wrong but the the math department of say Stony Brook or Indiana (ranked 25 and 34, resp.) is not smaller than that of Chicago or Columbia (ranked 5 and 9, resp.). Furthermore, the vast majority of professors in all of the places mentioned are alumni of Harvard/Princeton/Berkeley/MIT/Chicago/Stanford (i.e., a top school in my terminology); post-docs also come from very prestigious places to all of the four mentioned universities. So what makes Chicago or Columbia 'better' than Stony Brook or Indiana? Just their name?
 A: I have had a look at the rankings. While I think it's true that
math programs at the top of the list are stronger than those at
the bottom of the list, I think it is silly to say that number
$n$ is substantially better than number $n+10$ in any obvious sense.
Some departments are strong in some subfields and some in others,
and in a ways that can change every few years.
The direct answer to your first question is that courses will tend
to be more demanding and students will tend to be better at higher-ranked
departments. But if 'level' of instruction means quality of teaching,
that is not necessarily the case. 
However, those may not be the issues that are most relevant to you. If you
can get into one of the higher-ranked departments, you are probably used
to being at the top of most of your classes. And all of your fellow
students in the graduate courses can probably say the same. So you shouldn't
necessarily expect to be at the top of your graduate classes, no matter
how well-prepared you think you are. The competition may be an order of
magnitude greater than you are used to. Your willingness to work really
hard from the start and your motivation to keep that up for several years
will have a lot to do with your success.
With regard to your second question: In general, both the size and the quality of the department will affect the number and variety of new ideas you are
exposed to. It is certainly possible that you will find an interesting
and rewarding thesis topic at a small, lower rated department. But the
chances of starting a successful research career depend on finding 
one topic that fits your personal interests and capabilities. That will
depend to some extent on the size and quality of the faculty, but it will
also depend on your own interests and motivation, and also (to an extent) on
luck. 
One important issue you did not raise is how well the graduate department
matches your personal goals. 


*

*If you are mainly interested in an academic research
career, then a degree and with an excellent thesis from a top-rated 
department will be a real advantage in a very competitive job market. 

*If you are interested in an academic teaching career, then the quality of instruction
to which you are exposed and your own perceptiveness about the ingredients
in quality instruction may be the greatest predictors of your success.

*If you are interested in doing applied research in industry or government
agencies, then you should try to find a department where applied-oriented
research and consulting are valued, and with a track record of graduates
who have gotten jobs of the kind you want for yourself.
When deciding to which departments you will apply, you might use rankings
like the ones in your link as some indication about how competitive the
acceptance process is likely to be. However, you should do your own research
on the departments. Do not rely on a program website to tell you everything
you need to know. For example, you should consider what papers various faculty
members have actually published in recent years above listed 'fields of interest'. You should also try to visit any campus that offers you admission.
Does it seem to be a place where students are productive, motivated, mentored, and reasonably happy? 
For several years, I had a government job that
required me to visit many graduate mathematics departments. Some had much
better records producing successful and productive graduates than others. 
Although I was not fond of the travel, it seemed to me that this kind record was highly correlated with aspects of the 'atmosphere' of a department that were not observable except by a personal
visit.
