First off, I have issues in tegard to the rankings game. The lists-US News, Brennan, Princeton, etc- they can be helpful guides as to the general quality of the law schools. You should look at the individual programs instead.
If the rankings are credible at all, I believe they do not apply to LLM programs as the different law schools have their individual preferences-NYU/Georgetown, tax; Harvard, Yale, Stanford, government, regulation, etc, Columbia, interntational/corporate law. There is an LLM ranking link in this forum-the only one-where Columbia is number 1, Harvard is no. 2. But that, too must be taken with a grain of salt.
NYU is easier to get into than all the other so-called top law schools as NYU gets as many as 400 LLMs each year, two or three times more than any leading law school, but perhaps, not really easy compared to more accommodating schools. Yes, experience counts a lot since these schools want to have a diverse profile not only in terms nationality but more in terms of what you bring to the academic community since LLMs are graduate students. They are expected to contribute their experience to the intellectual discourse. And then there is the prestige part. They would rather take someone who has worked at say, the Rwanda Criminal Court or the Yugoslavia court over one who's wet behind the ears.
What I know is since they don't conduct interviews, the application letter and the recommendations carry so much weight, all things being, more or less, equal.