Floppy tapes are kind of the "budget" end of quarter-inch tapes. They can't be read by the higher-priced tape drives that use QIC02, QIC36 or SCSI interfaces. Essentially, they use the floppy controller to format, write and read back data in a "sort of" floppy format.
One aspect that can get nasty is that they don't do "read after write" verification. So if your tape has an error, you don't know it until to make another pass over the tape to verify it. And then, if there's an error, you have to start the process all over again. Whatever you do, don't get rid of the drive if you ever hope to recover the data that you've written--floppy tape formats aren't necessarily identical.
If you're really interested in quarter-inch tape, you might want to look at some of the SCSI-interface models. They're not expensive and they're pretty reliable, as long as the tape holds out. (Quarter-inch cartridges are beginning to develop age-related issues).
You can also find ISA floppy-tape "accelerator" cards that can transfer data at 1Mbit/sec. and don't tie up your floppy controller.