Johnny - I would be careful on trying to get them to match BU, or giving them the impression that is what you are after. I could see that getting their back-up, especially (in hjort's assessment) BU is a lower school.
You should ask, try and see if you can get something, but going form
nada to ask for a free-ride seems to stretch the hopes a little.
Obviously all the merit based chat is useful. Arguing it is more expensive to live in NYC? I can't see anyone falling for that, or yourself coming out great. Everyone knows NYC is expensive, so raising the issue could give the impression you don't really know what you are letting yourself in for by going there.
Although Admissions are now on the sell side rather than the buy, they are people you will interact with at school, and people involved in a lot of what goes on (there is a lot of student involvement in the Admissions Office as you are likely aware).
So, what am I trying to say? Not sure. 2nd round Stern has less scholarships, but some. So if you have zero now, asking them about it, or leveraging against BU, isn't likely to get you the near free ride at BU. It might get you something though, and is definitely worth making them aware of - I doubt it is something you have informed them of, and may sway them a little.