Like others here I chose an MSW due to its flexibility. I also understand that it's an older degree than counseling and has a bigger lobbying force behind it. In addition to being able to bill Medicare I think that most hospitals and the VA will only hire MSWs and it's just a more well recognized degree. What finalized my decision was looking at profiles of therapists and people who worked a therapeutic centers that interested me and seeing they were all LCSWs.
As far as NYU goes, I joined the accepted students FB page when I got in and saw threads of people who had graduated recommending one go elsewhere (Hunter) due to costs. I also spoke to a friend of mine who went to a private college (outside of NY) and said that her classes often had people straight from undergrad or otherwise with no experience or understanding of social work who ate up a lot of class time and that there were big problems with diversity in the student body in general. I believe a public institution will have a different caliber of student, especially one like Hunter which I have always heard is more selective than NYU, Columbia etc due to the high volume of applications they get. Like alittleshady said this is a super personal decision which in my opinion kind of boils down to are you wealthy enough to afford a private education. If so maybe it's worth it to you, but I've never seen anyone recommend to take out a ton of loans for an MSW.