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The only reason to use them is some form of conspicuous consumption, signaling how cool and wacky u areDisagree.
1. Sometimes the sonic characteristics of tapes DO complement the music. Typically for genres other than popular music or symphonic/classical/orchestral music (where absolute fidelity is king).
2. Physical media as opposed to digital media is, for some/many, inherently gratifying. It is a more multifaceted Thing, an object with a mass and volume and cover -- far more features than a typical audio file might possess. This point might not resonate with some on this forum (especially those interested in minimalism/efficient living), but ask around and you will find that for some people, this makes a dramatic difference.
3. Cassette tapes, as opposed to vinyl records, were extremely easy to produce in small quantities or at home (VHS tapes also have this property). This means that you can find many interesting/unusual cassette tapes (or make them yourself with relative ease). There are many recordings only available on cassette, and others very challenging to find on media other than cassette.
Especially in our world of information hyper-availability, there is a satisfaction in enjoying or creating a tape that only a couple people (or maybe even only one) have ever heard. Some might point to this as a core tenet of "hipsterism," but I argue this isn't about conspicuous consumption, it's about appreciation of novelty (which is a core aspect of... intelligence/pattern recognition capability) and history/unique narratives.
EDIT: looks like the article actually makes the same points that I did, but expressed differently and with a bit more contemporary context/narrative content.