In some scenarios you could use either, but it depends a bit on your use case and the UX you want.
The save to drive button is useful when you know the specific file that the user will want to save. Its a very simple UX where you just put the button next to the specific file(s).
The file picker is used when you want the user to select the file that they will update. Its a little more involved, but gives the users more control over what they select.