Button competed in F1 for 17 full seasons from 2000 to 2016, making 306 starts that included 15 wins and his world title in 2009.
The 40-year-old spent three of those seasons as Hamilton's team-mate at McLaren where he gained an insight into the six-time champion as both a driver and a person.
"I think he’s quite a shy character," said Button in an interview with GQ magazine. "You can probably see that. I don’t know Lewis now that well.
"Lewis in 2012 was an extremely talented driver, whose whole life was about racing. I had to work so hard to beat him, and obviously it didn’t happen all the time. But it happened enough that I was happy with what I achieved."
In their three seasons together Hamilton finished ahead of Button in the drivers' standings on two occasions, yet the latter out-scored his fellow Briton overall by 15 points - 672 to 657.
"On balance he out-qualified me, but I could race him, and use my head because it wasn’t all about driving flat-out in a grand prix," assessed Button.
"It isn't now, but Lewis has learnt a lot since then. He’s always looking after the tyres, the fuel, the car… he knows what he’s doing.
"He’s in a position now where he can relax and think about other things in addition to driving the car."
Button has also witnessed Hamilton evolve as a person, notably this year, with the 35-year-old dovetailing his pursuit of a record-equalling seventh world title with his push for greater equality and diversity.
"He can be very outspoken and I think it’s great that he is," added Button. "It will help the sport and the diversity of the sport.
"Whether or not you’re a fan of Lewis as a person, you should definitely be a fan of him as a driver."
In concluding about the 35-year-old, he said: "He is exceptionally talented, and he does not play games. He will never play dirty. In fact, I think he’s the cleanest guy I ever raced against."
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