My vague recollection is that the 720 KB floppy, a brief precursor to the 1.44 MB, was simply a floppy that used only one side of the disk, but at the same data density and track count, etc. In order to use them, just like any other drive, you first had to format them. And if you check out the options for the old DOS format command, you'll see that there are command-line switches to specify which type of disk you are formatting.
Now, modern disks come pre-formatted to the 1.44 MB layout. Put those into an old 720 KB drive and it can't read a format it does not understand. In fact, the drive lacks the second-side head that the disk's format info (in the MBB) wants to use. So you simply have to format the disk in the 720 KB drive to the 720 KB format, and it should work.
Now, the other way, if you have a formatted 720 KB disk with data and put it into a 1.44MB drive, it should recognize the format info and read it all just fine. Likewise, it should write to that disk, too, restricting itself to the lesser capacity layout by using only one head. that backward compatibility was designed into DOS, and I would hope it still lurks even in current Windows.