Cassettes are popular because it's the analog format for music that anyone can manufacture. All you need are some blanks and a deck, and you can start a label, right there and then. Beyond that, professional duplication services are still relatively inexpensive and can provide fast turn-around, generally.
Vinyl costs a lot of money up front, takes a very long time to make (lead times can be up to a year), and requires a certain amount of expertise to get a good quality product. None of those apply with cassette.
Also, not everyone cares about, or even desires, quality of sound. Degradation and imperfection as an aesthetic has a wide appeal - people seem to remain interested in the strange, mechanical world of analog music reproduction.