1) brand whoring. i went to a fancy-sounding undergrad, and it has actually been super helpful since graduation. The top schools really care about their reputation, and they put the time/$$ into maintaining it. Networking events, helping alums find jobs, etc. Not that NYU Law doesn't, but NYU-proper probably doesn't care about image as much as Columbia-proper.
2) NYU encourages a lot of people to go into public interest. Great. I don't care. I mean, I think it's an admirable thing, but I'm already going into law school with a public interest focus, and I don't think it helps me if a bunch of my classmates want to do the same thing as me. A bit selfish, but there you are. I kind of like the idea of being a striped fish in a polka dot pond... (yeah, i feel like i'm mixing analogies here).
3) Columbia has a bunch of faculty who focus on the stuff I'm interested in. NYU does too, but in a more decentralized, general way. This is the biggest plus, and is probably 80% of the reason I'm considering Columbia at all.
As an NYUer, I think that 1 and 3 are very good reasons to care about CLS, and 2 is true, but is more of a lack of a reason to pick NYU than an actual reason to pick either (don't forget that about 80% on NYUers are going in to private practice, despite all the PI pressure).
I would mount one small defense of NYU on count one. NYU undergrad is TTT, and everyone knows it. Strangers on the street know Columbia, and the prestige factor is real. BUT: NYU knows that Stearn and the Law School are the only things about the school that aren't an artsy joke, and they throw resources at both schools in acknowledgement of this. It won't change the name-recognition, but they do try and compensate by heaping extra help/resources on those schools.