Both sentences have the same number of hits on Google:
So I'm curious, then what's the difference? Or there's no difference at all?
She won a scholarship, so the didn't have to save up for her studies.
up there expresses the idea accumulate. Since save already has that meaning, the particle is just a kind of reinforcement of the idea.
After months spent in space travel in Zero-G, astronauts will need to build up muscle.
He is working odd jobs on Saturdays and saving up for a new bike.