A college applicant (well, his father) recently asked me:
"Is it worth it to spend double for private college over going somewhere public (in-state), given the difference in tuition?"
This is a question that I can't really answer for you - it's a personal decision that only you can make. If you could predict with 100% certainty that you'll go on to graduate school, then it would likely make sense to save the money on undergrad. However, it's difficult to predict whether you'll actually go on to graduate school either directly after undergrad or in the not-so-near future. Things change.
I went to Columbia for undergrad and thought I would go on to graduate school at various points (even filling out applications), but then didn't.
The "brand" of an Ivy League or other first-tier school carries a great deal of weight in a variety of areas. Pursuing any type of graduate school is less worthwhile than it used to be, so you may not end up going to graduate school in the end. Having a brand-name school on the resume (and to name-drop in various settings) is a valuable signal of intelligence/skills/aptitude, etc.
If the public university provides an equal education and the cost ends up being much less than the private school, there is no need to focus on name recognition.