Both Harvard and Vanderbilt have "need-blind" admissions. They accept students only based on their academic performance, test scores and, to some extent, extracurricular activities. They do not look at how rich or poor applicants' families may be to determine whether they will be accepted.
That said, there is some preference to "legacy" admissions (one or both parents attended the university). Legacy applicants still have to qualify for admissions, but they might not be quite as stellar as would be expected for regular applicants. (perhaps, for example, a 3.85 unweighted GPA and a 1500 SAT for a legacy; rather than a 4.0 unweighted GPA and 1560 SAT, for a Harvard applicant). While there's no guarantee that legacies are any richer than non-legacies, a long history of university-educated people in the family does suggest that the family can at least afford the tuition.
In any event, other top universities (for example, Tufts and George Washington University), have "need-aware" or "need-sensitive" admissions. They can't afford to potentially give every student financial aid, like Harvard can, so they accept enough students who can pay "full sticker price" to pave the way for those who need extensive financial aid. While it may seem like applicants that need no financial aid might have a bit of an edge over those that do, again, all the students admitted are fully qualified. These schools are very selective, so they're basically chosing rich students over middle-class students with identical GPAs and SAT scores. Instead of a coin toss, they choose the student who can pay full tuition, making more financial aid available for those who can't.
In George W. Bush's day, the legacy preference and family finances would get applicants a lot farther than it does today. Yale's need-blind admission policy only started (I believe) in 1964 (so his was probably the last class that had need-aware admissions), and there were far fewer applicants in those days.
Vanderbilt is very highly ranked. US News puts it as #14 (tied) in the U.S., equivalent to some Ivy League schools such as Brown and Cornell. Harvard is currently ranked as #2; it's nearly always ranked within the top three, alternating as #1 with Princeton, Yale, and sometimes Stanford.