Disclaimer: I’m a Rose alum and didn’t look at University of Cincinnati during my college search, so I’m much more knowledgeable about Rose than I am Cincinnati.
Two things that set Rose apart from other schools are its small size and undergraduate STEM focus. With the small size you get a familial atmosphere and more personal attention at the cost of not having things like a big sports program, a typical state college atmosphere, or a sprawling alumni base. With the STEM focus you get a student body that generally are all interested in STEM topics and have similar interests, and you get a more focused curriculum - at the expense of being more limited in non-STEM coursework and major options.
In general as far as programs go, my impression is that University of Cincinnati is a solid school but doesn’t particularly stand out amongst a crowd of good public engineering programs in the midwest (I don’t know anything about biomed in particular, though). Rose, however, has a unique education model and school that I think really does help give you a great undergraduate engineering education. I’ve taken undergraduate classes at both Rose and a well regarded public research university and I was surprised at how different it was, and how much the personal attention at Rose made a difference.
One of the main disadvantages to Rose is its cost, which sounds like it isn’t a problem for you. If you aren’t turned off by Rose’s small size and STEM focus, I think you’d really enjoy Rose and I would strongly consider it. I really liked my time at Rose and got a great education. If you’re looking for more objective metrics, I’d contact the career fairs at both schools and look at what companies recruit at the career fairs, what graduate schools students go to, etc., and see if the potential outcomes at either one draw you more than the other. I think Rose makes that information available on their website. Good luck with your choice! Either one will give you a great foundation.