What makes tape sound better than vinyl ?

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.

A flawed original can be copied to be made better in the same way photoshop can enhance a picture, but it is no longer an original. If that is your end game, you are better off getting an equalizer. 

I find cassettes generally sound very musical, rich and natural. By comparison CD generally sound thin and bland and "uninteresting." Oddly, perhaps, I find cassettes that are digitally remastered sound quite good - very analog and detailed with better dynamic range than their CD counterparts in many cases. Case in point - Kind of Blue digitally remastered on cassette is very dynamic, detailed and lush. It appears cassettes went out of style and production just about the time overly aggressive dynamic range compression reared it’s ugly head for CDs, later on for vinyl.