Hi, I'm a senior at East (East Chapel Hill High School), and I have friends that go to both other schools in the district, so I can tell you a little bit about them.
Carrboro has had a lot of racial tension within the student body since it opened, just as a result of the neighborhoods it is districted to, I think. I think you can probably Google this and find it in the news, because it was a pretty big deal when it happened.
Also, because Carrboro is newer, it does not have as many AP offerings, nor does it have a very established performing arts department (there is no theatre in the building yet. One will be added in a few years, I believe), so if your daughter is interested in theatre, chorus, orchestra, etc., East would definitely be a better fit.
Carrboro's building is nicer (and more environmentally-friendly), mainly because it is newer, but compared to Chapel Hill High School, East's building is very sparkly and new too.
Although the school system is trying very hard to convince families otherwise, I believe that going to a school with "Chapel Hill" in its name will help in the college admissions process, also. Admissions committees recognize the name and associate it with the (very real) rigor of our school system. If your daughter will be applying to an out-of-state school, this could be a factor to consider.
I definitely wouldn't say East is a country club (maybe compared to the 70's, prison-like architecture of Chapel Hill High), so don't be misled, hahah. The school is a giant building with a Hill on which kids eat lunch and throw frisbees. It is two-story, and you can look down on the lower level from the balcony-like hallway of the upper level, creating a nice open space. However, it's still a school. Linoleum, creaky lockers, and gum stuck on the bottoms of desks are all present.
Both are great, rigorous schools and I'm sure she will enjoy her experience regardless of where she goes- these are all little nuances of difference only visible to someone within the community. They are very similar schools in comparison with the rest of America's high schools.