A case could probably be made for either. Courant has an outstanding department; in particular it's one of the very best places in the world for applied math and analysis/pde's. As you're likely aware, the 2005 and 2007 Abel Prize winners are both on the faculty there. On the other hand, though there are faculty in other areas, the emphasis does swing toward analysis there, so if you want to avoid the word "differential" you might feel like a bit of an outsider. The financial aspect is a valid concern, and since New York is the most expensive city in the United States you'll probably have to borrow a lot in your first two years (I presume, though, that if you pass the exam and get financial aid it will be large enough to compensate for the cost of living for the remaining years). While big cities can be intimidating, quite a lot of people have fallen in love with New York after moving there, and Courant is located in a very desirable part of town. You certainly will have no need for a car, so traffic won't be an issue.

The departments in the 25-30 range are all certainly respectable, and I think each has some individual faculty members who are quite distinguished, but these departments are not of the same caliber as NYU's. They'll surely have the right infrastructure in terms of coursework that you need and so on.

If you don't already have this information, it might be a good idea to ask the graduate director at Courant what the success rate is for students who are admitted under a plan like yours, and how long on average they take to get their Ph. D. You should probably get similar information from the other school too. I'm not sure which will take longer--on the one hand, time-to-degree tends to increase as you go down the rankings (and it's also usually higher for public schools, but I don't know if your other school is public), but on the other your situation might make your time at Courant longer than it would be if you were admitted directly to the Ph. D. program.

At NYU, you'll be surrounded by a lot of very bright peers (and even brighter senior people), which has the advantage of possibly pushing you harder and creating a fertile environment for new ideas, but which also might feel intimidating and cause you to feel lost in the crowd. The other place might feel a little more comfortable. Part of your decision may come down to whether you think you need to feel challenged from the outside, as you would be at NYU, or if you're self-motivated enough that you don't need that outside pressure.

So those are some thoughts. I hope they've made you feel less confused rather than more so. Good luck in your decision.