Kone fired home a late winner for Wigan against Tiote's Newcastle at the weekend. The goal arrived after Maynor Figueroa appeared to help the ball on with his hand.
"I've known him for a long time but that's football," he told the club's website.
"When we are on the pitch, we're not friends but when the game is finished we can talk together.
"It was handball but the referee made a mistake. I think we deserved more but that's finished now.
"Everyone is still disappointed about the result. I thought we played well as a team but then we conceded a very sloppy goal which was handball, but the referee didn't see anything."
Tiote, like everyone at Newcastle, is still smarting from Callum McManaman's challenge on Massadio Haidara - which went unpunished at the time, and will not be dealt with retrospectively.
"I think that's a decision for the referee. Sometimes they make mistakes - it's not only the players - and we saw that with the tackle and the last goal, but he didn't see anything so that's bad for the team," he said.
"I think we did everything for the fans. We played as a team, we did well together and we tried everything to make them happy. We were all left disappointed by the result but it wasn't all our fault.
"It was really difficult for us because I think it was a really strong challenge on Massadio and I hope he will be back as soon as possible because we need him. He's a good player and I hope he will be okay."
Tiote says Newcastle need to forget the Wigan game and looked ahead to their next clash against champions Manchester City.
"It's finished now and we have nearly two weeks until we play against Man City, so I think we need to think about the next game," he said.
"It would be good to have another game straight away but that's football. Now we have no match for a week because we've got an international break. After that, we need to come back together and do everything to be ready for the game against City."