I think Columbia's public health program gets some more attention because Chelsea Clinton went there for her MPH, though she's been to a lot other private schools like NYU and Oxford, people like the idea of being in New York.
BU was ranked around 25th for years, they politic and that may have raised their US News rank, or there are NIH funds that overlap with public health, but rankings based on return of investment in a given school put BU around 25th,
Top 25 Campus MPH Programs. I wouldn't put a lot of faith in the rankings, these are all pre-COVID anyway. Strange things happened at BU with COVID that may have brought problems to light, a lot of faculty off the record disagreed with a return to classes, and one longtime professor at BU's school of public health, Dr. Michael Siegel, said that BU's "Learn from Anywhere" is more about making money (tuition) and less about health, and that is from his twitter from April 25th, 2021! A lot of BU employees come from the poorer Roxbury area and BU may have encouraged spread of covid in poorer areas, plus many at BU have asked people to join protests over the last year by declaring them safe from a public health standpoint, not sure if those asking for participation were actually at the protests, a lot of the danger comes from people meeting before and after protests indoors and a definite sum of protestors are in the ground now after getting covid or suffering longterm health effects.
BU just did weird things during COVID like cancelling summer abroad in London and charging students the same fee for Zoom classes as well as banning students from cafeterias and dorms but initially not refunding them for board and food! The name "Boston University" sounds like a venerable public institution, but it is more of a mix between expensive corporate higher education and a lot of advertising, many schools offer a more rigorous education with better connections like University of Washington and others . . . student satisfaction is low at BU. Public health school can be very expensive, what you get out of BU is more of a grab bag than anything, due to BU's budgetary problems (the whole university) they're actually at risk of having major future problems, I think it is better to go to a more firmly established public health school, BU's is the newest public health school out of many that are ranked high by US News, which is kind of suspicious, the area is super-saturated with public health schools and they may work to give each other high peer rankings as US News is biased in favor of schools in concentrated geographical locations and doesn't take value for the student into account.
On the flip side, Brown has had steadily worked for decades on their global health program and others and they bring an Ivy League comprehensive approach to a public health school that resulted in their applications skyrocketing the past year. Brown's dean Dr. Ashish Jha is very knowledgeable on global health issues (one of the people like Fauci and Walensky you know it pays to listen when they give an interview as their answers are concise yet well thought out and insightful) and correctly raised the importance of helping India during it's current covid peak which could be the worst this country faces . . . easily provokes tears if you watch what is happening in India now . . . I feel that Brown is very well positioned to work on achievable public health goals in the US and globally going forward. BU just seems super focused on a narrower range of topics and mostly domestically.