Intonation is a bitch.
Pure thirds, fifths, octaves sound so beautiful. This is what I found to be the most beautiful aspect of a string instrument yet. I'm convinced of this after listening to a descending third passage of Paganini's caprice. It just sounded so pure, so beautiful, so perfect.
So what makes the piano stand out? I've identified one aspect of what make the violin/viola stands out.
I think of the color when the pedal is depressed, like in Ravel's music. The lightness in French music, the etherealness of those music.
And the sound you can make on a piano that imitates bells. It truly reverberates and you are free to imagine anything that you want to associate with it.
Those are special effects.
Polyphony is another thing you pretty can only do on the piano. You can immerse yourself in your playing; but I'm not sure if listeners will be immersed as much.
Percussive passages are also quite effective--I think of Prokofiev's war sonatas.
I'm sure other things will come to mind given time.