For the random piano, you are right, there is no difference. For a paino being used to play a very specific piece, the tuning might be slightly different depending if the intended song is using D# or Eā, depending upon the key of the song. Though in such a case the piano might be tuned using a different standard that better fits the song.
One more extreme example is two pianos tuned to 12-TET, but one is half off. They are made to be played together by two closely in sync pianists for a few more complex songs that need 24 steps between octaves.
Overall I do find the system confusing enough to wonder if a better one won't one day catch on. And it might already have, I know some musicians who can't read sheet music but play by chords. It seems more limited in the level of detail you can specify, but works plenty well for the songs they want to play.
I worked as a harpsichord tuner during college, and this kind of tuning was generally only used when only string instruments were involved. Once a single instrument with holes, valves, or frets is involved, you have to use equal temperament. Almost nobody does specialty tuning.
One more extreme example is two pianos tuned to 12-TET, but one is half off. They are made to be played together by two closely in sync pianists for a few more complex songs that need 24 steps between octaves.
Overall I do find the system confusing enough to wonder if a better one won't one day catch on. And it might already have, I know some musicians who can't read sheet music but play by chords. It seems more limited in the level of detail you can specify, but works plenty well for the songs they want to play.