Band saw is well, a band (think belt, like the belts on a car engine) that runs on pulleys, so the saw moves continiously in one direction (downward). Plus the blade is much larger, for cutting large pieces. The blade is 1/2" or more from teeth to spine. ~~It's not really intended for making curves~~ it can do fairly large radius curves, and removes more wood because the blade is about twice as thick as a coping or scroll saw. A band saw couldn't produce the piecs in the post, because the saw band is a continuous loop.

A scroll saw is effectively a motorized coping saw, very fine, narrow blade, about 1/8" from teeth to spine, so it can do very short radius cuts. The blade moves up and down like a jig saw, but it's on a stationary unit - there's a deck you put your workpiece on and move the work on the deck around the blade.