Robert De Niro has been thinking in recent days about the concept of longevity. The actor has been in the business of making films so long – his debut on the big screen was in 1965 – that his work is now being restored.

"Yeah, I've been thinking about that. Restored, huh? It's kind of amazing," he says, sitting in his production hub in downtown Manhattan, the Tribeca Film Centre.

The restoration in question is the painstaking return to its original glory of King of Comedy, Martin Scorsese's dark 1982 satire on modern celebrity obsession, with its famous punchline: "Better to be king for a night, than schmuck for a lifetime." The movie has been digitally remastered from the original camera negatives and will be shown later this month on the closing night of this year's Tribeca film festival, with both De Niro and Scorsese in attendance.

"Rupert Pupkin, now what film was that one?" De Niro quips, feigning to have forgotten the part of the self-delusional would-be standup he played 30 years ago. "I haven't seen the movie in at least 20 years, and I want to see it – it will bring back memories not just of what I did in the movie, but of that period in my life," he says.

King of Comedy will be one of the highlights of the 2013 Tribeca film festival, the celebration of New York and its movie-making tradition that De Niro co-founded in 2001 when the dust of the fallen Twin Towers had barely settled over Ground Zero. The first festival, in 2002, was framed as a form of economic stimulus, the aim being to attract visitors back to lower Manhattan in the wake of the 9/11 attacks.

And it succeeded. Each year the festival has expanded, growing international tentacles until it can now claim to have screened more than 1,400 films in 80 countries, including through its Arabic offshoot, Doha-Tribeca. In the process, it has comfortably achieved its initial objective, generating about $750m (£488m) worth of economic activity for New York.

But therein lies a conundrum. New York City is back on its feet, as seen graphically in the form of the looming 1 World Trade Centre, the 1,776ft skyscraper nearing completion in Ground Zero just eight blocks away from De Niro's HQ.

The city's revival poses De Niro, together with his co-founder and producer of the Tribeca film festival Jane Rosenthal, with a fundamental question: in its 12th year, what is the festival's purpose now that its initial reason for being has been met and superseded?

De Niro says it remains his ambition to make the festival "part of the tradition of New York, part of the fabric; that I hope will be what it will be in years to come, and it's partially that now." His love of the city, and of its cinematic history, remains undiminished: "It's given me everything. I was born and raised in New York, I studied acting here; I love to travel, but I'm a New Yorker."

He might have added that the city also gave him his champion, fellow New Yorker Scorsese, who conjured many of De Niro's greatest performances in Mean Streets (1973), Taxi Driver (1976), New York, New York (1977), Raging Bull (1980), Goodfellas (1990), Cape Fear (1991) and Casino (1995).

De Niro recalls how he and "Marty" stumbled upon the look for Pupkin. "We were driving along Broadway, and we saw this shop with mannequins with flashy clothes like you'd find in places in Vegas. We jumped out of the cab and went in the store and I said to Marty, 'What do you think?' and he said, 'Yeah, let's take it.' And we took the whole thing off the mannequin, even the hairstyle."

Tribeca film festival's struggles with its purpose in life have been exacerbated by the proliferation of movie festivals around the world and the increasingly clogged nature of the annual calendar. Tribeca, dubbed "Hollywood on the Hudson", suffers from its timing so soon after the Oscars, which deprives it of the pre-Oscar buzz enjoyed by Venice, say, or Toronto.

Rosenthal accepts that this is a problem. "It does make things more difficult, and maybe if we have completed a certain mission then we would look to change the dates. It's been 12 years, so maybe it's time to think about it."

Like any festival, De Niro and Rosenthal are also having to grapple with the challenge of the internet and the opportunities it offers. This year they are extending across the Atlantic a films-on-demand experiment that has already proved successful in the US, exposing independent movies to a far greater audience.

The technique will be rolled out to the UK from 16 April, with an initial slate of six of the festival's films being offered for an eight-week run on pay-per-view through Virgin Media and digital platforms such as iTunes, PlayStation and Xbox. The selected titles include Greetings from Tim Buckley, a feature that explores the relationship between the father-and-son musicians Tim and Jeff Buckley; and Fresh Meat, a horror-comedy that profiles a gang of dysfunctional criminals who make the less-than-ideal decision to kidnap a family of cannibals.

As a further foray into the world of digital film-making at this year's festival, there will be a tieup with Twitter's Vine to launch a six-second film competition. This will be a competition for films lasting six seconds and posted through Twitter's #6secfilms hashtag.

Can that really be true? Robert De Niro, an actor whose attention to detail is legendary, lending his name to six-second films? "I think it's great," he protests. "It's pure image, and I like that. It seems to me a good exercise at finding a beginning, a middle and an end. I'm thinking, maybe I need to do it for myself."

• The Tribeca film festival opens on 17 April. Details: tribecafilm.com/festival