Villas-Boas, 33, joined Chelsea last month after a hugely impressive debut season saw him guide Porto to a treble-haul of Primeira Liga, Portuguese Cup and Europa League titles.
Mourinho achieved the same feat in his first full campaign at the Estadio do Dragao in 2002-03, but also went on to win another league championship and Champions League the following season.
And Porto chief Da Costa believes Villas-Boas' departure was motivated by a fear of failing where Mourinho succeeded.
"Villas-Boas was living with Mourinho's ghost," Da Costa was quoted as saying in the English press. "He has a complex, and is always thinking about the ghost of Mourinho - how Mourinho won the UEFA Cup and Champions League.
"He refused to continue here because he was afraid of not doing the same. He was afraid he would not have the same success as in his first season. I think that was the crucial aspect in his departure."
Villas-Boas has consistently moved to distance himself from Mourinho, who he worked under at Porto and followed to Chelsea and Inter Milan, but Da Costa believes the young coach made his latest career move too early.
"In a conversation [with Villas-Boas] I said, Are you worried about shipping three or four goals against FC Barcelona [in the Super Cup]? Mourinho shipped five but that does not mean he is no longer the best coach in the world.
"If I was his father I would have told him what I told him as his president and his friend: that he should have stayed at FC Porto. In football, when you talk about European giants, the measure is not Chelsea. It is Real Madrid, Barcelona, Manchester United.
"I told him that this would be Guardiola's last season at Barcelona and that was the ideal club for him, but he was scared of the comparisons with Mourinho and left."
Da Costa also appeared to suggest he would have no problem losing star striker Falcao to Villas-Boas and Chelsea, claiming that holding onto Hulk is his No. 1 priority.
"If 10 players leave, I have replacements for each one of them," Da Costa said. "There is only one who I don't have a replacement for - Hulk."