The single biggest thing you're missing is an attack. Pat Muchmore's comment is exactly right:
One thing that’s definitely missing: a real piano sounds has tons of non-harmonic components during the attack of the sound. The overtone series will help with the sustain and decay, but the attack needs lots of noise and inharmonic components.
Nearly all instruments have a brief, non-pitched, percussive sound that plays for a tiny fraction of a second when a note first starts playing. If you listen to your sample piano note again, you'll notice that the first split-second of the note is much louder than the rest of the note. You might also notice that it sounds a little bit like a snare drum is being struck when the note starts playing. That's the attack. (On an actual piano, this sound is caused by the movement of the hammer sticking the string.)
In contrast, your synthesized note is at its quietest when it starts playing. It has no attack. This missing attack is incredibly important! I believe that if you add an attack to your note, it will sound convincingly like a real piano.
You can do this by either playing a shaped burst of white noise when your note starts playing or, somewhat more convincingly, playing a sample of a percussion instrument, mixed in subtly, each time the piano plays a note. (Even though a white noise burst sounds pretty artificial, I've had very good success adding white noise attacks to synthesizers in my own music. It really is convincing enough to work in a lot of situations! Todd Wilcox's comment gives good advice on making white noise work for an attack.) The most convincing approach would be to find a recording of someone isolating the sound of a piano hammer striking without the sound of the piano string ringing and use that for your attack sample.
I did notice that your synthesized tone has a percussive element in it already. However, it sounds partway through the note, not at the beginning. The attack has to reach its peak at the precise moment the note starts playing. You added this graph to your post:
Notice that for the real piano tone on the left, the loudest point of the sound is all the way to the left edge of the sound wave. In contrast, your tone gets louder for half a second before reaching its loudest. Unfortunately, this is too late. The percussive sound needs to start at the beginning of the tone to be a convincing attack.
In addition, your percussive sound is very low-pitched. A piano typically has an attack that also has some mid-range and high-pitched elements. I suggest using a snare drum hit, soft wood block hit, or, as I mentioned above, a recording of a piano hammer strike to stimulate your attack instead of the deep handdrum sound you're using now.
Other than the attack, to my ear, your synthesized tone is almost there. I personally think that once you add an attack, you'll have a synthesized tone that's as high-quality as what you'll find in most $100 MIDI keyboards. It will still sound artificial, but it will sound pleasant and it won't seem strange.