To answer the OP's question, if anything the "top 20" schools are more competitive simply as a result of a numbers game. There are likely many more people that add the "top 20" schools to their lists as reaches because... why not? There's no harm in doing so short of the cost to do so, and who knows, maybe it'll work out. It's also difficult as the strongest applicants will typically apply to most or all of these institutions while they likely won't apply to many of the lesser ranked schools.
I don't think there's anything in particular that you "have" to have to get into a top 20 vs. a non-top 20. The reality is, though, that those schools have the ability to be pickier about the students they accept because the applicant pool is that much stronger.
My perceived difference would be:
-more likely to have gone to a better undergrad
-more likely to have done research
....And that's it. I know a few people that went to top 20 schools vs. top 40 schools and there is no clear difference in terms of intelligence or work ethic.
Top 40 vs. non-top 40: you start to see a little difference but both student bodies still have ranges that overlap quite a bit so it's still a little silly to extrapolate that to one individual.
Top 20 vs. under top 40: yes, the difference is noticeable. But there's still decent overlap even here.
There are definitely a lot of very smart people in my class, but I'd say the most striking thing to me is just how enthralled everyone seems to be with research. Many won't ever end up in academia but it still seems to be a pretty "academic" leaning bunch.
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