1.44Mb floppies have an extra cut-out at the 'user' end, similar to and opposite side to the write-protect tab, but without the user-settable sliding cover (that is: on the near right hand side looking at the top as you insert it into the drive). The purpose of this was to allow a micro-switch or optical sensor in the drive to detect automatically whether a floppy was 720Kb or 1.44Mb.
However, not all computer manufacturers chose to implement sensor-aware BIOSs, with the result that floppies could be cross-formatted, 720Kb as 1.44Mb and vice versa. If such a floppy was then placed in a PC that had implemented the sensing, it would be unreadable. †
As I haven't seen a 720KB floppy for sale in years, the only way to obtain one now for use in old equipment would seem to be deliberately cross-formatting a 1.44MB as 720Kb, but more modern equipment might not be able to read a cross-formatted floppy, making data transfer impossible.
The solution is to try fooling the sensor(s) in such modern equipment with a piece of opaque tape, for example electrical insulation tape, folded around the back edge covering both top and bottom of the RH cut-out, like so ...
|^ Direction of disk insertion >
However, electrical insulation tape itself is quite stretchy, so if the drive uses mechanical rather than optical sensors, something more robust may be required to avoid constant renewal.
And, of course, such pieces of tape sometimes come off in the drive, causing their own problems, so in the best IT tradition I accept no responsibility for the consequences of anything I say!
† This was a significant support time-waster with Compaqs, who were always ridiculously slow to implement such agreed standards - IIRC it was only introduced as late as their 486s, giving plenty of time for users to build up a stock of cross-formatted floppies. When we upgraded someone's 386, often the first thing that would happen is that many of their backup floppies became unreadable!