If you are looking to specialize - Stony Brook, hands down. From past graduates of both schools, Stony Brook offers a far better didactic education, smaller class size (and thus more personable letters of recommendation for specializing), and the school has a much better reputation than NYUCD. Not that reputation decides whether you can specialize, but you might be happier and more confident with your quality of education based on the years of experience of past graduates, which speaks volumes between the difference of both programs.
Although you can specialize from any school, it would be a large mistake to assume that the students at NYU will be pushovers in terms of working your way to the top. There are gunners at every school, even NYU. A good, hardworking student will be a good student in both a poor or rigorous program. Going to a less well regarded school won't necessarily help you stand out - just go to the best school you can.