One way to look at it is that the professional's repertoire is 30,000% or 40,000% larger than the amateur so they have tens of thousands of hours more development. A very advanced amateur's entire lifetime repertoire can be learned in under a single year or maybe two (and this may be overly conservative, probably just one year is enough) for a top professional like a Daniil Trifonov or Grigory Solokov.
So the phrasing and finer muscle control, sensitivity to the escapement, ability to fire the wrist, familiarity with wider ranges of motion with the elbow and more extreme wrist rotation, moving the torso, pedalling, ability to voice with the 4rth and 5th fingers trained in all sorts of double note textures and variations -> all those things will be learned and refined much more thoroughly as you go through the repertoire.
Even if a piece like the Chopin B flat minor nocturne is within a lot of amateurs' level, the professionai wll still sound better because he or she has performed most of the ballades, scherzi, polonaises, both concerti and both sonatas, a lot of the other nocturnes, waltzes, mazurkas, etc.
Tons of amateurs play the Moonlight sonata or Pathetique but the professional who's played 4 of the Beethoven concerti, and 20 of the other sonatas or even all 32 sonatas will still sound better.
A really advanced amateur might play something like the Liszt HR2 friska or La Campanella well but professional who has played the Liszt grande fantasie on la clochette de Paganini, El contrabandista, all the opera transcriptions, will perform it with a lot more freedom and bravura because they cut their technique on much harder jumps than in the easier Liszt, and because of the higher margin of safety they can divert their concentration more to color and clarity, and no longer have tendency to cover up flaws with heavy pedaling etc.
I suspect that psychological freedom to use the pedalling cleanly is a huge factor in this. If you can play a piece at a high level and clean up the pedalling - this alone and nothing else - will set you apart and allow listeners to wonder if you're professional.