The floppy disk hard-plastic case is protection for when you put the disk in your bag, pocket, etc. This is an issue we avoid with hard disks because they're not removeable.
The floppy disk drive needs to open the case:
Then the read-write arm can get access to the recording surface. A little hook grabs a hole in the cover down here and slides it over.
The recording surface inside looks like this:
So you see why it's "floppy". Mostly, it's much lighter. Also cheaper to mass-produce; I don't mind destroying one for a demo, or taking it to someone's house and just putting one little file on it.
See how that bit which the drive grips to spin it is attached to the media. There's a big hole in the back of the harder plastic case which we saw it peeking through before.
It's a little shiny. Not as much as the metal in the hard disk platters, but similarly we don't want any roughness or other material in the way of reading the surface -- we want it to be uniform.