My son did NYYS (the orchestra program, not the chamber program). It is more competitive and higher level than other city youth orchestras because it goes up to age 22 and also it is NYC and draws people in from LI, CT, NJ, Westchester, etc. He loved every moment of it and it truly inspired him. The violinist level is competitive but not crazy. I know of a couple violinists there that are good but not the best of their age groups and they are in the back of the 2nd violinist section. And then the front of the 1st violinist section is full of spectacular violinists.
My son is now at Juilliard PC and due to time commitment, dropped NYYS. Juilliard is all day on Saturday so there is no direct conflict (NYYS is Sunday) but he needs time to do homework and other commitments. That said, there are a lot of kids who do MSM PC and NYYS. Juilliard is hard to get into. Violin is not as hard has cello but it is harder than viola (in terms of getting in). I have known some kids who got into Juilliard PC who would not be able to gain admission to Juilliard College, and then there are the kids who are already recording artists who have already solo’d with major symphony orchestras. Violinists, because there are more of them, seem to have a wider range. Back in the day, there were so many more great violinists than cellists or violists, but nowadays there seem to be more violists (although still nowhere near the number of violinists) and way more cellists. As far as high school programs, this is about as competitive as it gets for classical musicians. There are, as someone said above, people who move to NYC from other countries to attend Juilliard PC so the competition involves not only NYers but also people from NJ, CT, PA, MA, I even know of kids commuting from VA, and then also the international community who also shows up to compete. I agree that it tends to help in college admissions at Ivies just like any other elite level extracurricular activity, but for music school is more about your audition and/or previously established relationships (like if you worked with a teacher at a music festival over the summer and he/she really liked you). Studying with a Juilliard teacher can help, but even if you don’t, you should reach out to at least one Juilliard PC teacher (this is goes for MSM as well) to have a trial lesson to see if you have good chemistry with them and also to hopefully impress them.