The following is a script from "Prime Minister Trudeau" which aired on March 6, 2016. Lara Logan is the correspondent. Max McClellan and Lani Levine, producers.

When Justin Trudeau comes to Washington later this week, Canada's new prime minister will become the first Canadian leader in almost two decades to be welcomed by the White House with a state dinner. It's an important relationship, the U.S. gets more oil from Canada than any other country and it's one of this nation's biggest trading partners. But relations lately have been a little frosty after years of Conservative leadership in Ottawa that was often at odds with the Obama administration. That changed when 44-year-old Trudeau took office last fall. His father, Prime Minister Pierre Trudeau famously made Canada one of the most progressive countries in the world and many in Canada wonder if Justin Trudeau would ever have made it to the country's top office without the most storied name in Canadian politics.

Canadian Prime Minister Justin Trudeau CBS News

Lara Logan: You've had a somewhat unusual path to this office of prime minister.

Justin Trudeau: Well, I was a snowboard instructor, I was a bouncer in a nightclub, I was a whitewater river guide for many years. I worked as a teacher. I make no apologies for a very varied set of life experiences.

Lara Logan: But it's also opened you up to criticism. I mean, we've heard it, you've heard it. You know what people say, that you're too young, you're inexperienced, that, you don't have what it takes to do this job.

Justin Trudeau: Well, I-- the way I respond to it is by ignoring it I mean, you cannot let yourself be defined by the hopes that you will fulfill the darkest wishes of your opponents.

"Well, I was a snowboard instructor, I was a bouncer in a nightclub, I was a whitewater river guide for many years. I worked as a teacher. I make no apologies for a very varied set of life experiences."

Justin Trudeau's sweeping victory was not expected.

[Justin Trudeau: This is what positive politics can do!!]

A few weeks earlier, his Liberal party was last in the polls...yet when the votes were counted, he'd done what no other leader in Canada had ever done: taken his party from its worst defeat in the last election to an historic win, snatching power from the Conservatives after nearly a decade of rule.

In that moment of victory, his youth, his looks and his family name captivated the world.

Lara Logan: Is that daunting?

Justin Trudeau: It is what it is. I look at what I have as a challenge and I could list a whole bunch of different challenges. And I choose not to be daunted by any of them.

Undaunted and still untested, with a majority government that gives him significant power he says he'll use to return the country to its liberal roots.

He's already fulfilled one of the boldest promises of his campaign: welcoming 25,000 Syrian refugees, some of them in person...

[Justin Trudeau: Welcome to your new home.]

...at a time when the U.S. has taken in a little over 2,000 refugees from Syria and governments are more concerned than ever about security risks.

Justin Trudeau: We were able to actually go and pick and choose and screen and bring over the people we chose. And that gives us a much greater level of control and attention over who's actually going to come in.

Lara Logan: But are you saying there's no risk? Or do you acknowledge that there is still a risk?

Justin Trudeau: Every time, every time a tourist or an immigrant or a refugee shows up in another country there's a security risk. And, I am more than comfortable that doing what we've done, accepting in 25,000 Syrian refugees does right by both the safety of Canadians and by the values that define us as a nation.

Lara Logan: Would you be just as comfortable if there was a terrorist attack carried out by someone who came through as a refugee?

Justin Trudeau: Ultimately, being open and respectful towards each other is much more powerful as a way to diffuse hatred and anger than, you know, layering on, you know, big walls and oppressive policies.

His commitment to openness is reminiscent of his father, Pierre, who's regarded as one of the founders of modern Canada.

[Pierre Trudeau: Canada must be progressive, and Canada must be a just society.]

He enshrined into law a charter of rights and freedoms -- similar to the U.S.'s Bill of Rights - that still defines what it means to be Canadian today and he made the country officially bilingual, giving French the same status as English.

Known as much for his towering intellect, as his glamour and charm, he dominated Canadian politics for nearly two decades.

And, from the moment his son was born on Christmas day in 1971, Justin Trudeau was thrust into his father's spotlight -- every step, including some of his first, chronicled by a nation obsessed.

His dad took him all over the world to meet popes and prime ministers and royalty.....his little brothers sometimes at his side.

[Reporter: What did you think of Lady Diana?

Young Justin: Oh she was very beautiful, And I'm glad that prince Charles has picked her.]

It was a unique childhood, but it defined him in Canada as Pierre Trudeau's son and he struggled to break free of that legacy.

Lara Logan: You still face the people who say, "This guy didn't earn it. You know, he's trading on his father's glorious past."

Justin Trudeau: I'm proud to be his son. And I don't mind that people remember that. I think that's a good thing. But one of the things that comes with that is having lived all my life with people who would criticize me without knowing me because they didn't know my father. Or people who loved me without knowing me because they loved my father.

Lara Logan: And both are false.

Justin Trudeau: Both are false.

In 2012, four years into his political career, he chose an unusual way to prove he was more than the spoiled son of Canadian royalty. He turned an annual charity boxing match into a political opportunity, challenging, Patrick Brazeau, a senator from the opposition who had a black belt in karate.

Canadians took one look at the two of them, and said Trudeau had lost his mind. The first round went as expected. But he and his trainer Ali Nestor, had prepared for this battle. He would take a beating in the ring, tire his opponent and outlast him.

Sophie Trudeau: I think I was like this. Looking through the fingers. It was not good.

Trudeau's wife Sophie was in the crowd.

Sophie Trudeau: You're thinking, why is he doing this again? And I don't like seeing this. But, he told me so many times. "I got this. It's not gonna be easy. I got it. I can do it."

Lara Logan: Did you know the strategy going in?

Sophie Trudeau: I did know that, but he admitted that the punches were with such strength and force, he had been punched before but not with that strength. And there were some moments where, you know, he was seeing stars. But he stayed upright.

Trudeau held firm to the plan and when Brazeau tired, pounded him into submission. The referee had to step in before the end of the third round.

Justin Trudeau: People think that boxing is all about how hard you can hit your opponent. It's not. Boxing is about how hard a hit you can take and keep going. That ultimately is much more the measure of a person-- than someone who says, "Oh, I've never been knocked down," or, "I've never been punched in the face." Well, you know what? Maybe you should have. You might learn a few things about yourself.

The prime minister still boxes at the same gym in Montreal where he and Ali trained for the fight. We stopped by to watch them work out. The kids here all know him -- and still call him by his first name. With his triumph in the ring, Trudeau proved he was tougher than most people thought, a strength that came in part from a life defined not just by privilege, but by tragedy.

His mother, a media sensation nearly 30 years younger than her husband, struggled with mental illness -- then undiagnosed. When she left her husband and her boys, the painful separation played out in the tabloids. And when Trudeau's younger brother Michel was killed in an avalanche in 1998, the loss was very public. Trudeau says his father was never the same.

Pierre Trudeau died two years later. It was the largest state funeral in their history and more than 20 million Canadians watched Justin Trudeau -- then 28 -- deliver the eulogy for his father.

Lara Logan: That was a moment that had lasting impact. Why do you think that was?

Justin Trudeau: I denied this for a long time. But I think there was a sense of showing people what a great father he was by showing what love his sons were capable of giving him.

Today, it's his family people come to see. His children growing up in the spotlight, unphased by cameras and temperatures well below zero at the popular winter carnival in Quebec City, where we joined them in a ritual he and his father used to enjoy.

Much of his time as prime minister is spent here on Parliament Hill in Ottawa, the country's capital. This is what they call the lifeblood of Canada's democracy, where laws are made and Trudeau responds in public to questions from the opposition. The hallways we walked with him are filled with his memories. His father lay in state here and his portrait hangs on the wall, a constant reminder of the man he has to follow.

Lara Logan: How long have you been in politics? About eight years?

Justin Trudeau: About eight years. Yeah.

Lara Logan: And you went from zero to prime minister for...in about eight years?

Justin Trudeau: Things fell into place and there was an opportunity for fresh leadership. And-- I was-- I was successful.

Under his leadership, Canada is redefining its role in the world. He pulled the country's fighter jets out of the U.S.-led air war against ISIS, but more than doubled the number of advisers on the ground.

Lara Logan: Your role in this fight is bigger now than it was with just six planes in the sky.

Justin Trudeau: There's a lotta countries that do very well at dropping bombs. There are other things that Canada actually does better than most other countries. And one of them is training people on the ground.

Lara Logan: But it's not disengagement. In fact, it's a -

Justin Trudeau: No.

Lara Logan: --deeper engagement in the war--

Justin Trudeau: Indeed.

Trudeau's father liked to say that hockey players and cold fronts were Canada's main exports. But the U.S. relies on its northern neighbor for more than that. The State Department says more than eight million Americans depend on Canada for their jobs and nearly $2 billion in trade flows across the border every day.

On the eve of his visit to the White House, we asked the Prime Minister about Canadians' feelings toward the U.S. and were surprised at his candor.

"So having a little more of an awareness of what's going on in the rest of the world, I think, is what many Canadians would hope for Americans."

Lara Logan: What do Canadians not like about the U.S?

Justin Trudeau: You know, I had a conversation one time with an American parent of a friend of mine and she was a big supporter of a presidential candidate. And I pointed out that if that person was run-- if indeed this man was running to be-- as Americans like to say, the "most powerful man in the world," I just felt like it might be nice if they paid a little more attention to the world. So having a little more of an awareness of what's going on in the rest of the world, I think, is what many Canadians would hope for Americans. Because you can't be Canadian without being aware of at least one other country, the United States, 'cause it's so important to us. I think we sometimes like to think that, you know, Americans will pay attention to us from time to time, too.

Editor's Note: A previous version of this story included a photo inaccurately identified as Margaret Trudeau. The error has been corrected.