If I were to live with only one it would be hands down a bandsaw. For interior cuts it's simple to cut in from the outside and do the cutting then pass back out and close the gap with a sliver of wood glued in the saw kerf. And for that small added step I get a saw which cuts smooth as a laser instead of shaking up and down like a paint mixer and trying to shake the work around.
The downside is that most bandsaws do not have a deep throat. That's where a scroll saw really shines.
As it happens I've got both a band saw and a scroll saw. I keep trying the scroll saw to see if I can tune it or try a different blade but in the end it's all for nothing and I go over and hug my bandsaw after getting frustrated with the scroll saw every time.