None of us on are on the committees for these things, so you probably won't get a completely accurate response. However, I think most of the top 10 schools have some baseline qualities they want from students. This is typically the "first filter". Now moving beyond that you still have a large pool of extremely qualified students. You have students who were top of their class, perfect GPA(or at the very least 3.9+), perfect exam scores, research experience, publications, math competition prize winners,IMO winners etcetera... Now the last few attributes(e.g. Putnam,IMO, ...) are icing on the cake but they aren't necessary to get into a top 10 school. However, the first few attributes do demonstrate that the student is diligent, can handle graduate coursework, and can focus on working on a project for long durations(i.e. the thesis).
Once these attributes are reviewed you pass a "second filter" so to speak. This is where admissions depends on the committee. The above attributes will also be carefully reviewed along with your letters of recommendations. There's lots of people with good grades and good test scores, so this is where letters of recommendations come into play. Most generic ones(i.e. did good in my class, etcetera...) don't distinguish the candidate from any other one, so if the letters says your a strong student and have great potential as a researcher then that's where it gets interesting. If you notice Princeton has admits with relatively low GPAs being admitted, that's because those students typically have publications already and are already working at a high level with people in their fields. Private schools will have really high standards like these. The public schools(Berkeley, UCLA, Michigan, etcetera...) on the other hand take a more holistic view and balance out these attributes. There are many students at these public schools that came in with more standard curriculums and no publications. The students who got admitted that didn't have one of the first few attributes (high GPA,high test scores, etcetera...) usually had some extenuating life circumstances that created that situation, and again this is where letters of recommendations and the statement of purpose will become important. There are also other factors such as being a minority which is also factored into decisions at these public institutions, but that's a bit less clear what they do in those cases.
At the end of the day the committee just wants reliable candidates that can do well and finish their program. They are making a pretty long term investment(5 years, $125000) in you after all.