WIth regard to utility in admissions and pursuing medicine/dental/pharmacy/podiatry/optometry/chiropractic/etc...
I personally think private UNDERGRAD is useless for the health profession admissions. However, for law, business, etc. being an alumn of a private school will make a big difference. Every LARGE university will have the basic science classes to teach you what you need to know for the DAT and health prof admissions. The difference comes from the perks. Does your state school provide an extensive research network? How about excellent resources for you to use at your liesure, like facilities. Is the school convenient and conducive to your learning and success? Everyone needs different, but I'm one who can adapt to all institutions, so I'd take a state school over private for undergrad (if you have a choice). In fact, I'd have taken my state dental school over Tufts if I weren't rejected (I'm still bitter at UMDNJ for their blatant lying), but everything works out in your favor in the end.
If you foresee that you're definitely going to attend a health prof school after undergrad AND you are very resourceful with and laid back about what you have, save the cash and take the state school
What you should worry about is what programs at your state school suck compared to your potential private universities, and vice versa. Sometimes the private school will have better research opportunities. For example, Stanford will kick the crap out of most (if not all) Cal State Universities. On the otherhand, Rutgers University (I feel) spanks Princeton in terms of health professions preparation. I will, though, take Princeton over Rutgers any day when it comes to pre-law or pre-business.
HTH.