What makes tape sound better than vinyl ?

nmps, i think you have to accept that we all hear differently. There are characteristics of analog reproduction are a more pleasing to people, just as there are characteristics of digital reproduction that are pleasing to others.

I am also in a business that overlaps with the sound business and did an analog v. digital test on a feature film mixing stage. We did a mix and recording of the final sound of a film (of which the music score is a separate element) to hard drive as is the norm now.... this on the finest equipment available, directly from a Pro Tools session. We also simultaneously recorded to 6-track magnetic film which is analogous to reel to reel and the way film sound used to be mastered . We did a playback of both units so we were able to A/B and there wasn't a person in the room that didn't think the analog master sounded way better. More real. More human. With the music especially, the analog master showed way better accuracy of tone and reproduced the sense of the room the music was recorded in way better than the digital master. I think there's just something about analog that reproduces ambient space and it makes tone and space feel more real.

 "We also simultaneously recorded to 6-track magnetic film which is analogous to reel to reel"

This is not cassette and in no way reflects the inherent problems with cassette.

The point isn't about whether analog or digital is better but, rather, whether cassette or high rez digital is better. And, on that note, IMO, there is no contest. High rez digital wins out by far.

"Cost factor and maintainability are not necessarily convincing arguments when dealing with audiophiles. You know..... "

For ultimate sound, you are right. But, for the cost you would need to spend on cassettes you would be better off going with reel to reel. 

Further, to demonstrate the issues with cassette compare 1st generation through 5th generation copies of a cassette duplication. All the glaring problems and distortion associated with cassettes are laid bare! Couple that with not using noise reduction, as some have advocated, and the problem is even greater.

I've owned many of the best cassette decks and still own a Nak 700 mkII and Kyocera D811. I have even better decks in the past. My experience on location, in the studio have easily led me away from cassettes many, many years ago. However, I still have hundreds of hours of masters that I keep for archival purposes. All of which have also been transfered to 96/24 using the best digital equipment!

Back home and able to hook up the Nak 582 for the first time.
To start with I had no idea it was so LARGE!
I have it connected in my second system as it is too wide to fit in the rack of my main system.
Impressions so far are very positive, first few tapes spun through sound very lively and airy.
Quite impressive for $100......serviced 5 years ago and then put away. My steal of the month.

Ive heard recordings made from vinyl to an Akai GX something? Reel to reel. I know this isn't a Tandberg or Revox, but I didn't think it sounded better than vinyl. I noticed losses, increased hiss, loss of top end, and less bass. 
But I know what the OP is talking about. Tape has a certain sound.
I take objection to his assertion that tape sounds "better". I agree it sounds different, but don't agree it's better. Back in the days of tape and before CD, I didn't hear anyone saying a recording of vinyl on tape sounds better than vinyl. To me this assertion is weird. I can understand the OP having his personal preference, as we all do, but I don't agree with it.  If he simply asserted that he enjoys the sound of tape, I'd be OK with that. 

But, to somewhat answer his question. "What makes tape sound like tape", in my experience from recording live music on it, and recording records and CDs on it, I'd say it's the losses and technical limitations that make it sound like that. Those losses and limitations round off transients and blur sounds together, and this gives a certain sort of sound. I know a guy who has a studio who has on occasion mixed down his multi track digital recordings to tape to get "that" sound as an effect. I agree that it can be a pleasing effect when used artistically. All formats to my ears have a intrinsic sound of thier own. To my ears digital has the most detail and least colourations. Vinyl though pleasant has a lot of resonance and noise when played back - listen on headphones if you want to hear it easily. Tape, is second best to digital, it has a dragging smeared and grainy sound compared to digital. On its own, tape can sound nice.

Any unremasterd CD of tape recorded music (meaning any music recorded from 50's to late 70's)  reveals the characteristic sound of tape. 

Compare those recordings to any music digitally recorded and the absence of smear, noise, grain and colourations is immediately apparent when hearing the digital recording.

Alot of discussion here on this topic, i got reel and cassette and cd and lp, to make a point? All i can say with certainty is  at chicago axpona 2017 show, i went to alot of rooms and listened to some uber expensive turntables , but one small room i visited had the best sound ive ever heard in my life, Frank Zappa was playing on a seriously good sounding/looking Technics 10" reel playing 15" p.s. probably a master tape, my God! i could close my eyes and reached out and touched the man! All those 30-100k turntables sounded pretty mundane compared to that, so in my mind and ears the RTR tape @ 15 ips cannot be beat providing the deck and tape are 1st rate in this life or the next to come,imho.