There is malaria and Dengue in Belize in certain areas. There is Lyme's Desease on the east coast of the US. There is Rocky Mountain tick fever in the rockies, Swine Flu is everywhere. The point is that is you let that all control your decisions, you would stay home and never go out.
CC is beautiful, although a bit laid back for my tastes. AC is a more active. Both are really safe if you use basic common sense.
If you have children, I can not imagine a better place to vacation with fun for all age groups. I even had my 89 year old Dad here for a while and he loved the bird watching and all the attention that the lovely Belizean people showed him.