I agree with the above answer that you should go to the college that is the best fit for you and where you can excel. Take the time to visit and meet people at the colleges you think you are interested in. It's a very personal decision and money can and usually is a factor. However, I wouldn't discount a college because of its sticker price. You might be eligible for non-loan based aid, you might be awarded merit scholarships. The point is that a college that is more expensive on paper might not be once you apply and get accepted. You'll never know until you try. If then you find that you can't swing the cost then at least you are making an informed decision. There are colleges where the tuition is very reasonable, but they offer less aid or scholarships and you might end up paying more out of pocket in the end. So, I would suggest applying to both if you can.
As far as name recognition, I can only speak indirectly for the study of medicine but I know in the study of law, if you go to a school with less of a "name" it becomes more important that you do well. I've heard people say in law that there is no real difference between the best student at a name (or Ivy or top ten) school and the best student at a second tier school. However, things being the way they are, an average to above average student from Yale might have an easier time than an average to above average student from a different school. But a failing student from any school is a problem. And some schools have a reputation that -- if you get in, you'll graduate with a 3.0. Where other schools might admit more students and then only the best students rise to the top. If you aspire to teaching, like being a professor at a university or at a teaching hospital, the name would likely matter.
But again, the bottom line is if you are happy and comfortable where you go to school, you will likely do well, and that will take you very far. Good luck!