I would say that it really depends on the program and the courses that you take.
I would agree with Senior007 on the fact that it depends on the courses.
For example, an MPH concentrating in Epidemiology is different from an MHA
BUT and MPH in health policy & management can be identical/similar to the MHA.. That is to say, an individual with an MPH in health policy & management can do the exact same thing and can possibly qualify for any job that a person with an MHA can do and vice versa.
There are people who have only an MPH who are presidents, CEOs, etc. of hospitals and other healthcare organizations. UC-Berkeley and Yale offer MPH programs that allow you to be a healthcare executive with an MPH
My friend's cousin graduated from Emory University with an MPH in health policy & management and she got a health administrative fellowship with Emory Hospital. So, I think that a person with an MPH in health policy & management can do everything and anything a person with an MHA can do. You have to also remember that MHA people perform statistics just like teh MPH people do. Make sure you check wiht the programs and look at teh curriculum to make sure there are business classes (Economics, strategic capability, technology, HR, etc.) If a couple fo the classes are missing, take some for electives and make sure that the people in the program are interested in you being successful and assisting you with that.