Spurs would have moved ahead of Chelsea into third if they had secured a win against the Potters, but ended the afternoon in fifth after both Arsenal and Everton secured victories against Wigan and West Ham respectively.
Villas-Boas, whose side face Aston Villa on Boxing Day, was not happy at seeing his team slip below Arsenal on goal difference.
"No one enjoys being level on points with Arsenal at this club," Villas-Boas said.
"I have experienced this rivalry for the first time this year and I have spoken very highly about it before.
"It is an extremely healthy competition between both teams, but we have to continue now.
"Today was a difficult game. Stoke are an extremely competitive and physical team who are difficult to break down.
"I want to be extremely positive as the players really tried hard."