Full ride at Duke, absolutely no question. Prestige is, at best, a bad way to choose almost any school, and even in that, there is no real difference - while Dartmouth has the “Ivy” label, Duke has more name recognition. In any case, even if Dartmouth WAS more prestigious, it is not $24,400 more prestigious.
Duke is more diverse and has more poor and minority students. While Dartmouth has a higher Pride index, and better residence solutions, Duke is considered much safer a place for LGBTQIA people.