The 2008 Formula One world champion has been an enthusiastic user of the medium, acquiring some 1.2 million followers. Some of his more trigger-happy 'Tweets' have generated plenty of news coverage for all the wrong reasons, however.
In Belgium last month, Hamilton raised eyebrows with crude language and then sent out confidential team technical data hours before the race. Button said at the time he was 'disappointed'.
The morning after last Sunday's Japanese Grand Prix, he accused Button of having snubbed him before retracting his comment once it emerged the older Briton had never followed him in the first place.
The messages triggered plenty of media debate, much of it analysing Hamilton's state of mind and maturity as he prepares for a move from McLaren - the only team he has ever driven for in F1 - to Mercedes next season.
"I just have to put my hands up and say I made a mistake. Sometimes we do that in the heat of the moment and you don't actually mean what you say, but there is nothing I can do. I apologised and that is it, we move forward," he told reporters when asked about the latest incident.
"Perhaps me and Twitter don't go together too well," continued the 27-year-old ahead of Sunday's Korean Grand Prix.
"I like it because I like to be able to share the great experiences I have with the fans but it is a fine line. Lots of people get it wrong and I am one of those and I will hopefully try to improve on that."
Hamilton said he had not been able to contact Button directly by telephone on the Monday because his team mate had a late night after finishing fourth in the race at Suzuka. They had spoken again in Korea on Wednesday.
"I obviously wasn't in the right frame of mind, but that's it. Hopefully I won't do it again," added the Briton, who has not tweeted any messages since apologising to Button on Monday.
Button, somewhat bemused to be talking about a 140 character message format rather than his car or Sunday's grand prix, said he did not follow his team mate because he saw him at every race weekend.
"The fun thing about Twitter is you don't always follow the same people, you can mix and match, follow and unfollow," he explained.
The 2009 champion does follow Mexican Sergio Perez, who joins McLaren next season in place of Hamilton. Button said he had done so to welcome him to the team but could not now 'unfollow' him.
"I'm not allowed to do anything on Twitter these days. I'm stuck. I can't unfollow anyone any more," he said.