Though he signed the German to Ferrari, by the time Vettel arrived at Maranello in 2015, di Montezemolo had departed the Italian company.
With four successive titles to his name, much was expected of the German, and as Ferrari sought to recover from a relatively lean period, Vettel saw the opportunity to emulate his great hero, Michael Schumacher, by pulling the team together and returning it to title-winning glory.
However, a combination of factors, not least internal politics, the team's habit of shooting itself in the foot and Vettel's own 'red mist moments', mean that thus far, much like his predecessor, the German has only managed runner-up.
Last season, in particular, in the latter stages of the season Vettel appeared to be struggling, making countless unnecessary mistakes, be it sliding off the track while leading at his home race or numerous first lap kamikaze bids that totally compromised the rest of his afternoon.
Away from the track the German seemed down, no longer enjoying his racing, and though he looked strong again in pre-season testing, he was clearly shocked by Ferrari's poor pace in the season opener.
Montezemolo suggests that Vettel will never emulate his hero Schumacher, believing that the German lacks the necessary mental strength.
"My feeling from outside, and from outside sometimes you are wrong," he tells the official F1 website, "is that he's quick, a nice person, he loves Ferrari, he knows the responsibility.
"But I think that maybe he doesn't have the strength sometimes," admitted the Italian.
As previously reported, Montezemolo also questions the mentality of Vettel's predecessor, Fernando Alonso.
"To be honest, he was unlucky to be in Ferrari when Red Bull was the same, or very close to being, as Ferrari at the beginning of the 2000s," admits Montezemolo. "Second, I don't like to but I have to say this, he was unlucky. He lost the championship in Brazil in 2012 when Sebastian in the first lap crashed with Senna. Or, when the team made a big mistake in 2010, when it was enough to finish fourth, and we lose the championship at the last race.
"Third, his character. The biggest difference between Michael, and also Niki, is he was Alonso, not Alonso with Ferrari. When he wins, he's happy. When he doesn't win, it's the problem of the team, he's unhappy. He's less close to the team than Michael and Niki, particularly in the difficult moments. To be close to the team in the good moments is easy.
"Alonso in the race is a fantastic driver, was, and is," he admits. "He lost the championships not for his mistakes. In normal conditions, without the mistakes of the team is 2010, and without Vettel very, very lucky in the last race.
"If he won one championship, today we're talking about something different," he adds. "We're talking that Alonso, despite for many years he hasn't got the best car, he was able to win the championship."