Do universities care about what school you come from?

Currently, I don't go to a great school and admissions at top schools are quite low, However, I have the choice to change schools for my last 2 years to a better one, but I am a bit hesitant to leave since I am quite comfortable, but I don't want to stay if it makes a big difference. My current school does not give much extracurricular activity choices or any big opportunities that may help and I want to take medicine which I think makes it harder to get into a better university.