Both are great schools, but I think it's a little unfair to UT to discount its great programs because it's a public school. Vanderbilt has a name, but so does the University of Texas.
As far as opportunities....
UT Law - 14th, Vandy - 16th
UT Business - 17th, Vandy - 28th
UT Engineering, for undergrads, is one of the top programs in the country.
Texas is an academic and a social powerhouse. Highly ranked programs, a diverse student body, a great college experience (football, etc.), and, of course, 6th Street.
Not to mention, for premeds, you have a plethora of good in-state med schools. Baylor and UT-Southwestern are great.
Just to clarify, I'm not knocking on Vandy, but Texas is a great place to be an undergraduate. And, you could always go get that private school name for medical school (I graduated from Texas, and I'm headed to a top-20 med school up north).