OP, I graduated in 2006 from UMiami with a degree in BME and am currently at UM med finishing up my second year.
Unlike some of the other posters before me (it really is an individual opinion kind of thing), I really loved majoring in BME and would not have done anything differently if I had to repeat it all over again. I have three very good friends who majored in neuroscience so I actually know quite a bit about both programs at Miami. Both majors will prepare you equally well for medical school. Choosing which one to go into really boils down to two things: which one you think you will enjoy more, and which one of the two will give you a career you can enjoy and do for the rest of your life if for whatever reason you don't go to medical school. I picked engineering because I found it fascinating (do some research on what the entire profession of BME encompasses and you will be blown away) and because straight out of high school I wasn't 100% sure I would go into medicine. BME provides you with an excellent alternate career, even with just a BS. An MS will really open more doors/$$$ for you and there is a combined BS/MS program available in which you can complete both degrees in 5 years instead of 4+2.
If you follow the "pre-med" route, both majors will have you taking all of the same required classes for medical school: biology, chemistry, organic, etc. Neuroscience is through the school of arts and sciences and in this major you will have to complete a greater number of social science and humanities electives than in BME. It's part of the curriculum. Additionally you will be taking extra higher level courses in the biology department (like endocrinology, physiology, developmental biology, etc.) as well as several major-specific classes dealing directly with neuroscience. Most students in neuro get involved in undergraduate research of some type at the medical school at the latest by fourth year and this can help you make contacts and meet people. It is a great major to prepare you for medical school. However, if you do NOT go to med school, a BS in neuroscience is pretty worthless. In order to make a half-decent living (and do anything but petty lab-slave work or teach high school bio for the rest of your life) you will pretty much have to get a PhD and prepare for a career in research. The PhD can take anywhere from 4-7 years....that's how it is.
Engineering is more intense in that it is not just a major but actually a professional program of sorts, like architecture for example. It's preparing you for a job in the industry straight out of school pretty much. That being said the curriculum has fewer of the extra required humanities and SS classes to make room for the extra classes in BME. Words of warning: the program is extremely heavy in math and physics. That's how it is. If you love these subjects and do well in them in high school, you should not be worried. This really depends on the individual Some people hate it, others love it. Keep in mind you will have many, many courses that do not DIRECTLY apply to a future career as a doctor. You will spend many hours in design labs playing with electrical circuits, writing often very frustrating programs in matlab (sometimes pulling your hair out), taking physics WITH calculus instead of without, doing higher math like differential equations, signal processing, etc. You also have to do a year-long senior design project in which you essentially design, plan, construct, test and "market" either a new device or research study under the guidance of a faculty mentor.
Is the program time-consuming and difficult? Yes, but very doable. It is NOT pre-med suicide as some people say. My graduating class of BME in '06 had about 30-something students, and about 10 of us were pre-med. Five of us (myself included) are now second years at UM med. Another guy went to FSU, another three went to medical schools out of state and only one guy didn't make it and stayed in engineering (he also happened to not take the mcat seriously and blew it off). That's 90% that got into med school in the first try. That's pretty damn good. I managed to get almost a 3.9 GPA, crush the mcat, maintain several EC, shadow docs fairly regularly AND maintain a very active social life at the same time. I enjoyed the program and would recommend it if you think you may be interested. Again it varies depending on individual preference. I would do it again if I were back in high school.
If you have any specific questions, please feel free to PM me. If you'd like I can give you a tour of the campus and facilities and introduce you to the dept. faculty so you can see at least see what the place is like. The only thing is I probably won't be free until mid-june (I will be studying for the USMLE Step 1), but let me know. On a final note, congrats on UM!! Any major you choose will be great and you will love the campus...the girls are amazing too...you'll be blown away 😀