Son has friends with experience in all these programs other than Rutgers. With the caveat that this is second or third hand (and perhaps a little out of date), some impressions:
Juilliard - smallest of the programs, about 25 undergrads in total, and extremely selective. May only have one opening per instrument in any given year. Quality of students and faculty very high. I’ve found the Juilliard jazz ensembles to be a cut above anywhere else. Can sometimes be a stressful environment, but nothing like the “Whiplash” reputation. My son has found the jazz students/grads to be collaborative (jazz is an intrinsically collaborative genre). Juilliard name can also create opportunities, both while in school and thereafter. If you can get in, hard to turn down, assuming cost side works out.
MSM - also very high quality, some amazing musicians and strong faculty. Bigger program than Juilliard, but still relatively small and selective. There are a lot of MSM students/grads on the NYC jazz scene (of course, Covid has largely put the NYC jazz scene on pause, but it will come back). Not located in the heart of NYC jazz venues, but some good places to play in the area and easy access to the Village by subway.
New School - generally considered among the top three jazz programs in NYC, with MSM and Juilliard. Has a reputation for being less structured, so it relies on self-motivation. Extensive adjunct faculty list allows students with self-motivation to study with some of the top names in jazz. Also possible to slack off if the motivation isn’t there. Located in the Village near some of the top jazz clubs.
NYU - I’m only aware of one of son’s jazz friends who went there. He enjoyed it but ended up taking a path away from music after graduating. And I don’t see many NYU grads in the NYC jazz scene. Don’t know if that’s just me though.
CCNY - flies under the radar a little, but the director of the program, Steve Wilson, is great. Son’s friend there studied saxophone with Wilson and had an excellent experience. Don’t know the cost, but I would expect significantly less than the other NYC schools. Definitely worth trying to gather more info.
William Paterson and SUNY Purchase - I don’t know that much about these two, but both have good faculty and there are regularly people from both in the NYC jazz scene and they sound good. NYC is reasonably accessible from both schools (in pre-Covid days and presumably again once we get to post-Covid), though of course not the same as being in the city. I understand both are relative bargains.
I’d also make the point that so much of success is a function of inner drive. All of these programs provide an environment where a student with enough inner drive can be successful. I’ve also seen some great young musicians in the NYC scene who went to non-name brand schools (e.g., Portland State). Not that the school doesn’t matter, but the main goal is to find a place where the student will maintain his/her drive and love of making music.