The scroll saw is for lighter duty work and you have much more finesse, greater detail, and you can saw in all directions while moving the work. Not to mention you can start from a drilled hole in the work.
A band saw is a big step up the duty ladder. You can cut much thicker pieces faster and still do much the same thing. Except for starting from a drilled hole and cutting in all directions.
I was in the same boat as you for several weeks until I found this baby for 40 bucks on CL. Was in a family for 50 years according to the original owner.
Attachment 125923
I put it on a temp stand to see if it ran and WoW. The more I played with it the more I liked it. It uses all the blades (3" to 6 1/2" ) ever made and is smooth as can be while running. I haven't tried yet but I bet it would take a piece of band saw blade up to 1/4" . They don't make em like they used to and after a little restore work I am proud of the purchase.
Kind of like the razors we love so much. Bought an old one cleaned it up and use it.
So I would say it all depends on what all you ever want to do with it. A good shop should have both but not everybody has the room or need for that. A scroll saw will do more than some would think but you have to let the tool do the work. It may take 4 times as long to cut a brass plate 1/8" thick but this particular scroll saw it will do it. Maybe thicker if the band saw blade works out.