Injuin
Where do students typically live? The cost of living in NYC is atrocious. Are there nearby towns that students commute from that have people calling them "Stern towns"?
What would you say sets Stern apart from Columbia?
Students typically live around campus. However, this depends on whether you already have family/partners. There are a lot of people who commute from Jersey or stay away from campus in one of the Burrows.
I agree, the cost of living is atrocious. But you are paying for Access! This has been the most important tool for networking. At Stern, you are more connected to the business world than any other school.
For your last question, I will tell you two things for now - The NYU Stern application, interview and post-admit communication was more personalized than any other school. For Stern, you are NOT just another admit. Also, Stern places more emphasis on the EQ (i know it sounds like a stupid term). This EQ is nothing both how social, friendly, down-to-earth and normal you are.