As every audiophile will acknowlege, compact cassettes have a sharper audio fidelity than any other widespread format.
Jokes aside, I really enjoy having a library of the music I enjoy that I can tangibly hold. I have an utter disdain for streaming corporations, and digital music doesn't appeal to me (though I do have a pretty large downloaded MP3 and FLAC library on my laptop and thumb drives). I do use spotify on occasion, but have been trying to make an effort to move away from it entirely. While I also collect vinyl records and CDs, cassettes are my preferred format for a variety of reasons.
I generally prefer analog formats to digital formats; while I think the whole "warmth" thing is a little silly, I'm not bothered by the fuzzy sound quality (my hearing isn't really good enough to distinguish it most of the time anyway). I also find the encoding methods used in analog formats much more personally interesting. Tapes are a lot cheaper than records, and have the added bonus of recordability. I love making mixtapes, and appreciate the ability to record an album that would otherwise be pretty expensive to acquire. Plus, I greatly enjoy the actual recording process. As you can probably tell from the rest of my site, my punk DIY ethos is really important to me.
There's also the aspect of portability; throwing my walkman and a few tapes in my bag is a lot more convenient than having to lug my entire stereo setup and records somewhere. It works for DJs, but I'm just one guy walking his dog or going to the store.