What makes tape sound better than vinyl ?

I don't find that to be true for cassettes vs vinyl.

I use well-tuned Nakamichi 682ZX with custom transformer with Maxell Vertex tapes. Recordings from cds also sound better than cds, in the same aspects. Cassette is just a mini reel, four track, slow speed. But of course I meant mostly open reel decks. Still, it's the tape.

No.
There is no way to improve a given signal.
It can be possible that the tape has higher gain in the output stage and this can give the impession of better dynamics / detail in a given stereo system.

I can run the phono thru the deck, this gives extra gain at the slight expense of the resolution. The turntable has better dynamics and detail and soundstage, tape sounds more natural.

Tape was an imperfect medium that degraded the sound. You may like that particular flavour, but it cannot possibly be an improvement if accuracy is the criterion. As a demonstration that it is indeed an alteration (and hence degradation), tape sound can be emulated very accurately in plug ins. See here, for example: https://www.waves.com/plugins/j37-tape#butch-vig-billy-bush-j37
Another way to investigate this is to do muliple passes of the copying: make a copy of a copy of a copy. After a number of generations there is nothing left worth listening to.