MHA is similar to an MBA except specific to healthcare, it teaches you how to manage a health organization like a nursing home, hospital, and physician practice. They have MHA programs, MBA with healthcare concentration, and dual MHA/MBA programs, they all end in pretty much the same boat, = management/administration.
A MPH will focus more on broad policy to help large health issues, they typically do large level statistics analysis. A large focus in MPH is Epidemiology. Epidemiology is the science that studies the patterns, causes, and effects of health and disease conditions in defined populations. It is the cornerstone of public health, and informs policy decisions and evidence-based practice by identifying risk factors for disease and targets for preventive healthcare. They also have MPH with health admin/management tracks.
if you pursue health admin, make sure it is a CAHME accredited program, stay away from online programs. I cant speak for MPH, but I know the salaries in general for public health are not as high as administration. If you go to a CAHME accredited MHA program with a decent reputation, after graduation pursuing a hospital fellowship you can expect 50-60k avg starting, If you start a real job right away 70-80k starting (hard to do unless you had previous experience. Pay is mainly dependent on the size of the hospital and cost of living of the area you are in. A 100 bed hospital is going to have salaries much less than a 800 bed hospital. Salary grows with experience. The average size hospital ceo makes well north of 250-300k a year. Much higher in large systems like mayo, emory, john hopkins, etc, 7 figures+.
Feel free to pm me if you have questions