A bandsaw can re-saw a thick board into thinner boards. A scroll saw can't.
A scroll saw blade can be threaded through a hole to make inside cuts. A bandsaw blade is a continuous loop, so it won't work the same way. You can cut in from the outside with a bandsaw, then slide the blade out, and finally glue the thin gap back together. Because the bandsaw has such a thin kerf, the glued cut can be nearly invisible.
Both saws cut curves, but the scroll saw seems better suited for intricate, delicate, detailed work. Both saws can cut your fingers, too. I imagine that a scroll saw is safer, but they are both generally considered much safer than a table saw.
(I was taught that bandsaws got their start in the meat industry. According to my bandsaw expert friend, bandsaws are always trying to get back to their roots when they see your fingers, so be careful!)