MOTOR SPORT NEWS - In his new autobiography, Life at the Limit, the 37-year-old - who was world champion in 2009 - describes Hamilton as "a brilliant, mercurial driver".
But he says the 32-year-old was "unpredictable" and he "regretted that despite our similarities, we were never really friends".
Button praises Hamilton for "really coming on over the past few years".
He added: "He's matured, become a bit of a statesman and a great representative of the sport."
Button, who retired at the end of last season but made a one-off return to Formula 1 at the Monaco Grand Prix this year, says he will never race again in F1.
He says the death in 2014 of his father, with whom he was very close and who accompanied him to nearly all his races, had taken the fun out of it.
"I've been offered drives in F1 but I'm not interested, although I will race in the future," Button says. "Just not F1. Not without dad."
A difficult relationship
Button describes three-time world champion Hamilton as "one of the greats", adding: "Of everybody on the grid, he's the guy who really has that 'gift'."
But he said their personal relationship was marked by tension throughout their three years as McLaren team-mates from 2010-12.
Button says he was warned by several people about joining McLaren because of Hamilton's talent and the way the team was perceived to be focused on him.
Button believes his arrival at McLaren with his entourage of close friends and his father changed the atmosphere, and that Hamilton feared it undermined his position with the team.
Describing their relationship early in their first season together, Button says: "Personally, he was fine with me, no issues at all at this stage of the game, but you could just tell he was a little bit peeved.