I don’t, well, for a long time I didn’t because it was a storage space issue. I’ve thrifted for movies and shows and put the discs in those big disc booklets that can be hard to carry. But at the same time they kept things in one place than having to play around with cases and whatnot.
However, I did decide a while ago that I will keep the cases for tv shows because I already filled up one big disc book. And now I’m deciding that for movies that I care about and would want the case for, they stay. Any movie that I don’t care as much for, they get stored in the disc book and the case is donated.
That’s another thing too is I was stupid to throw away the cases, knowing better that some places or people could actually use them. Worse off with the art cover included, which people could use that. Not anymore.
Music, yes; movies no.
For various reasons, I like having the “entire package” of a CD or cassette (though I don’t have many of those any more, just a few local bands that no longer exist). Not just the little booklet, but the covers too. I tried using one of the “books” that holds the discs and paper materials, but it never worked right. It’s fine for stuff that didn’t have any kind of covers or whatever, and CD singles, but I like the experience of opening the case, using the disc and having the materials there all in one package.
Vinyl is totally different thing to begin with.
Movies, I ditched individual cases a long time ago. The only ones I keep are special editions and whatnot.