the answer to your question will take days to answer. It might even be better to ask admissions, faculty, alumni, and current students.
At one point, I was considering those schools and Hopkins for HPM.
Here is a high high hiiiigh level overview of my thoughts on the management track of the HPM programs at those respective schools.
Yale - take classes at SOM (benefits of taking business classes with business students), amazing faculty, international focus (if you so desire), one of the strongest programs for public health at Yale (in my opinion of course). Alumni connection is amazing and extremely great with internship search between 1st and 2nd year. Good mix of students coming directly from school and with years of work experience. A lot of opportunities to get involved with research projects (if you want).
Columbia - umm... wasn't so impressed. I personally love NYC but I just couldn't get myself to like the program. The program coordinator informed me that for the 50-60 students, there were only 5-6 research positions available and was difficult to work with professors. and she was rude.
Emory - great atmosphere, amazing hospitality, beautiful buildings, yummy food, etc. I liked Atlanta and I thought they did a stellar job with the 2-day open house. Big class (I believe 35-40 students just in management and another 30-40 in policy). It's a big school and there are definitely pluses (infinite alumni network) and minuses for that. I think Emory spends/gives a lot of money on/to students, and they really make an effort to find you a job upon graduation. You have a lot of opportunities to work even in school, especially at the CDC since it's right across the street. BUT for me, I thought the students lacked work experience and for me that was important. I know that they are all brilliant students but for health management, I see it similar to business school in the sense that I value experience and marrying that experience into the classroom experience.
I hope that helps.
Again, this is my sloppy HIGH overview of my opinions and what I considered during my visits. And I did visit all three schools.