I think I might have some insight. I'm a sophomore in the Computer Science Department at UIUC, and my college choice came down to the Econ program and Chicago, or here.
The only thing that matters for the colleges you mentioned is what you want to study. Public or private doesn't make as much of a different as the strength of the department. It would have been silly to study Econ at UIUC instead of Chicago, and vis versa for Computer Science. If Computer Science is what you want to study, you can't go wrong with UIUC, Berkeley, or CMU. If you want to study Math on the other hand, you should definitely give Chicago serious consideration.
Besides the strengths of the departments, there is no different in terms of education. Just because CMU is a private school doesn't mean it's not a research institution. Same with Chicago. You may get a slightly better student to faculty ratio at a private school, but if you are in a top ranked department you will run into many professors who simply think teaching is a waste of their time and you will still have classes taught by TAs. You will need to be a motivated, independent learner, no matter what. You wont have anyone holding your hand. If you want a more interactive educational experience, I highly recommend checking out some of the top liberal arts schools like Amherst.
This is not to say that you can't get to know professors at research schools, because you can. You just need to work hard in order to do so. This is as opposed to Amherst, for example, where the vast majority of professors are there because they _want_ to interact with students. At CMU or UIUC, the vast majority of professors are there to do research. If you do go the research school route, you'll just need to try harder and find the professors that do care. It's definitely doable.
Of course, I have only addressed a few academic aspects about choosing a college. You should put equal weight on where you feel the most comfortable, where you think you would be the happiest. For example, if drinking really makes you uncomfortable, you'll need to recognize that Greek life at UIUC is huge. Similarly if you are worried about being in a large city, you might want to consider that when thinking about going to Chicago. All of these things are just as important to consider as academics. It is four years of your life, after all =).
You can post back here, or my emails in my profile if you'd like to know more.
EDIT: Something I forgot to mention is money. Public schools have none of it. If you actually need money from financial aid, go to a private school that guarantees to meet 100% of demonstrated need (all competitive private schools do).
The only thing that matters for the colleges you mentioned is what you want to study. Public or private doesn't make as much of a different as the strength of the department. It would have been silly to study Econ at UIUC instead of Chicago, and vis versa for Computer Science. If Computer Science is what you want to study, you can't go wrong with UIUC, Berkeley, or CMU. If you want to study Math on the other hand, you should definitely give Chicago serious consideration.
Besides the strengths of the departments, there is no different in terms of education. Just because CMU is a private school doesn't mean it's not a research institution. Same with Chicago. You may get a slightly better student to faculty ratio at a private school, but if you are in a top ranked department you will run into many professors who simply think teaching is a waste of their time and you will still have classes taught by TAs. You will need to be a motivated, independent learner, no matter what. You wont have anyone holding your hand. If you want a more interactive educational experience, I highly recommend checking out some of the top liberal arts schools like Amherst.
This is not to say that you can't get to know professors at research schools, because you can. You just need to work hard in order to do so. This is as opposed to Amherst, for example, where the vast majority of professors are there because they _want_ to interact with students. At CMU or UIUC, the vast majority of professors are there to do research. If you do go the research school route, you'll just need to try harder and find the professors that do care. It's definitely doable.
Of course, I have only addressed a few academic aspects about choosing a college. You should put equal weight on where you feel the most comfortable, where you think you would be the happiest. For example, if drinking really makes you uncomfortable, you'll need to recognize that Greek life at UIUC is huge. Similarly if you are worried about being in a large city, you might want to consider that when thinking about going to Chicago. All of these things are just as important to consider as academics. It is four years of your life, after all =).
You can post back here, or my emails in my profile if you'd like to know more.
EDIT: Something I forgot to mention is money. Public schools have none of it. If you actually need money from financial aid, go to a private school that guarantees to meet 100% of demonstrated need (all competitive private schools do).