What wonderful and very different choices !
Regarding both schools, has your daughter taken the time to look through the course catalogs and reviewed what the requirements look like and does one school stand out.
IMO, one advantage to Barnard is that your daughter can take courses at Columbia (outside of her requirements) and that makes the catalog and options quite large and extensive. The visiting professors at Columbia can also be a draw for many students. I love the Barnard Modes of Thinking required courses and the connection of so many courses to the city/real world.
The difference in living expenses will also be larger - NYC is quite expensive and while NYC does offer a lot of “free/inexpensive” fun, it can be an issue for students without some spending money.
Your daughter must be an excellent student to be admitted into Barnard RD and had strong essays around the desire to attend a women’s college. There are a mix of students that spend most of their time at Barnard and others that get involved with the broader community. Clubs are mixed between schools, dining halls open to all etc -so it is a way to get a small LAC experience and also the benefits of a large research institution, with the resources available at Columbia too.
At the end of the day, you are living in NYC with all the pros and cons that come with urban living and that is not the right place for everyone.