Yes, brand name matters - though I think the difference between NYU and Columbia, especially within education, is relatively miniscule. I might be wrong however. There's a lot of discussions about name value within academia if you want to join the conversation: https://chroniclevitae.com/news/929-academia-s-1-percent
In a doctoral application, recommendation letters and research experiences often carry a lot of weight. TC - while it admits a lot of students and usually doesn't fund them, hence the diploma mill reputation - has professors that are the top of their field. Having a well-known professor vouch for you can greatly benefit you. The trick is to not get lost in these programs - which is entirely possible, considering there is 100+ students in the masters program - and might actually be worse than attending a lesser known school.
Ultimately, I think you should look at fit, money (and for me, institutional resources) when it comes to NYU and Columbia. For International Education, there's various subfields you can specialize in - literary education, peace education, education policy, education in emergency, monitoring and evaluation, human rights, environmental education, assessments, curriculum & teaching, etc - and you want to make sure there's classes being offered in that field while you're there. You don't want to go too much into debt - so factor in scholarships, work opportunities, etc. And look at institutional resources - I had the opportunity to organize panel discussions on inequality and educational opportunity with some of the top economists / political scientists / education scholars while at Columbia, which I found to be a hugely rewarding experience.