A better guide would be COA placement rates, since a far higher percent of each school's class goes on to those clerkships. My sense is that NYU is edging Chicago out in COA placement these days -- particularly on the Second Circuit. A quick glance at recent hiring data shows why: NYU has placed at least fifty five clerks on the Second Circuit since 2007. Chicago, by my data, has placed only one. Things are less lopsided on other prominent circuits. NYU leads on the D.C. Circuit, thirty three to fifteen since 2004, but Chicago leads twenty seven to nineteen on the Ninth and twenty one to three on the Chicago-based Seventh. This data is largely self-reported, and so subject to inaccuracies/incompleteness concerns. Still, I think it shows that NYU has a slight edge at the COA level.
I'll note that using these numbers, Chicago does significantly better on a per capita basis. That's 110 NYU clerks and 66 Chicago clerks, but the class sizes are approximate 450 and 190. Dividing by the class sizes, you get .244 and .347. So Chicago students are about 42% more likely to get a COA clerkship on a per capita basis.
I have no idea whether these numbers are even remotely representative of the true clerkship placement numbers, by the way, both in the given circuits and outside. Also, in my experience, NYU students seem somewhat more likely to clerk at the USDC level, especially in the more competitive districts (SDNY and DDC). There's a bit of a "COA or bust" mentality for many Chicago students.
ETA: I was also curious about a few other circuits that I have experience with/friends who clerked at -- 5th, 8th and 10th. Based on GTL's (very helpful, even if incomplete) numbers, in these three circuits, there are 29 Chicago clerks, and nine NYU clerks. Throwing those into the 2/7/9/DC numbers, that brings us to 119 NYU clerks, and 95 Chicago clerks. Normalizing for class size, and you get .264 and .500.
Though now I'm curious about the First and Fourth, where I'd suspect more NYU students. Let's see... Either eight or nine NYU clerks (one was 2006-07; not sure if GTL is including that), to none for Chicago. So adding those in, you get 127 NYU and 95 Chicago. Normalizing... .282 and the .500.
So, I think the bottom line here is that there are going to be about 30-40% more NYU clerks on a raw basis, but 50%+ (and probably more like 70%) more Chicago clerks on a per capita basis.