Button is better at deciding when to go on the slicks at the optimum time for a drying track.
Most drivers either go to early or too late.
You only get that with experience and also some amount of care freeness, which only Button has.
Barichello is the only other driver i see who does this consistently, and it's no coincidence he's in the over 200 GP club.
If anyone notice as well Button was spinning the tyres out of the pits and down to turn 1 to heat them up. He's a tricky son of a gun, don't expect him to let the other side of the garage know that.

Hamilton doesn't have anything to learn from Button really. Team just made a poor choice. It has nothing to do with Maturity either. He's a proven winner when the team gets it right.
As for Alonso, he makes just as much mistakes. Made a couple in Hungary as well.
But it's all down to trying to get the car to the front. The ferrari is not good in the wet.
He's not as extreme these days as marcush says. He's still as forceful as ever though.
What i do realize with Hamilton though is that his wet driving is rather interesting.
He doesn't drive curved lines in the wet. He drives in TRON like angles, which seem to give him very good traction.
We can see this in the opening laps of Silvstone. It takes real confindence in oversteer behavior to snap the car into thos sharp angles.
It's the same reason he didn't run of into turn 2? in Hungary, he just didn't open up the steering like the others.