I guess it all boils down to what you want to do with your life. And I'm not talking about what you think is best, but on what you really wish you could do.
If you want to help people and you have a vocation be it in teaching or in medicine, a degree is needed, but if it just plain success without actually doing something specific, I guess only your wits can save you.
I want to get into research or work on something along the lines, so bioengineering is what I think would help me more. I'm a scientist by nature... I need a degree to get far, although the school wouldn't matter much, as long as it's a good school... in the end, your work will talk for you, right?