Just crunch the numbers. NYU tuition is about $3000 less than Columbia. But, NYU 2bdrm housing is about $45000, while student housing at Columbia is an "average" of about $30000 for a 2bdrm. That means it is about $12,000 cheaper to go to Columbia (for a 2bdrm) than NYU (based on 12-month occupancy, the difference would be less pro-rata for 9 months).

So, is it worth $12,000/year to live in Greenwich Village, rather than Morningside Heights?

Even if Columbia were more expensive than NYU, it is worth going to Columbia!!!! You can take the train and go to bars and restaurants downtown... In matters of education, Columbia is better than NYU for sure and Morningside Heights is also a great place to live, you will have a complete school experience and study in one of the best schools in the U.S.!