There’s a world of difference between a work that gives us what we didn’t know we needed and one that merely fulfills expectations. Expectations are just a thing we bring with us, as undistinguished and uninteresting as an empty water bottle; the gifts we haven’t thought to expect are the secret sauce.

A Beautiful Day in the Neighborhood, directed by Marielle Heller and starring Tom Hanks as the cherished children’s television host Fred Rogers, is one of those movies you didn’t know you needed. It’s drawn, to a degree, from the friendship between Rogers and journalist Tom Junod, who profiled Rogers for Esquire magazine in 1998 and found himself surprised by the man’s depth and genuine warmth. And if you saw last year’s marvelous documentary by Morgan Neville, Won’t You Be My Neighbor?, you’ll already have some sense of Rogers, an ordained minister and a longtime Pittsburgh resident who loved swimming and playing the piano; he’d been a chubby kid, taunted by his schoolmates, and he never forgot how that felt. Neville’s documentary showed how closely Fred Rogers hewed to his TV persona, in his kindness and his strong desire to help children to express their feelings—although, as that film and Heller’s both point out, he was a human being, not a saint.

That’s how Hanks plays him, in a performance that’s more a gentle watercolor re-creation than an act of overt mimicry, which would be the easier route. A Beautiful Day in the Neighborhood opens just as the TV show, Mister Rogers’ Neighborhood, did, with Hanks-as-Rogers opening the door to a small, comfortably furnished house. It’s really a set, of course, but you believe it’s a house, a refuge from the world equipped with all the things—a toy trolley, a kingdom of talking puppets—that help Rogers address the sometimes perplexing realities of kid life. Once inside, he switches his suit jacket for a zip-up cardigan, and trades his daytime loafers for sneakers. He shows us a board dotted with doors, like an oversized advent calendar. Behind each door is a picture: There’s one of King Friday XIII, one of Rogers’s puppet characters, and another of Lady Aberlin, the gentle soul who would often appear on the show to help Mr. Rogers guide children through their complicated feelings.

But behind the third door, looking like the subject of a Weegee photo after a street brawl, is a stunned-looking middle-aged guy with a bloody gash across his face. In his cosmically soothing voice, Hanks-as-Rogers introduces this poor schmoe as his friend Lloyd Vogel, and as he does so, his expression shifts to a mode of pure empathy, as easily readable yet as infinitely layered with meaning as a Kabuki mask. “Someone has hurt my friend Lloyd,” he says, “and not just on his face. He’s having a hard time forgiving the person who hurt him.”

This is how Heller and the movie’s writers, Micah Fitzerman-Blue and Noah Harpster, segue into the story that takes place outside of Mr. Rogers’ safehouse, in the larger world. Lloyd, played by Matthew Rhys, is a successful journalist and new father, married to a woman who clearly loves him, Susan Kelechi Watson’s Andrea. It’s been years since he saw or spoke with his father, who abandoned the family at a crucial time. A wedding brings them together; their encounter ends in a fistfight. The rest of A Beautiful Day in the Neighborhood shows how Lloyd ultimately makes peace with his father—played by Chris Cooper, conveying the current of melancholy that so often underlies human sourness—in a progression that’s realistically one step forward, three steps back. It also shows how Lloyd meets Fred Rogers, a man who will change his life, by reluctantly accepting an assignment from his editor. He’s a serious journalist, not a writer of puff pieces. Mr. Rogers is too weightless a subject for him, or so he thinks.

A Beautiful Day in the Neighborhood shouldn’t work. It’s one thing to have feelings, but who wants to see a movie about them? But Heller is a marvelous filmmaker, and the movie she’s made is radically, subtly inventive. Her 2018 Can You Ever Forgive Me?—starring Melissa McCarthy as iconoclastic writer Lee Israel, disgraced after she was caught selling forged letters, ostensibly written by famous literary figures, to dealers around New York City—was one of the finest pictures of that year, a funny and sympathetic portrait of a classic Manhattan misanthrope: One who doesn’t really hate people but who generally finds them wanting, and thus avoids them rather than feign interest.

Heller is just as observant here. Just when you think A Beautiful Day is veering toward sentimentality, she and her actors steer it toward more muscular, and usually more unnerving, emotions. Rhys gives a finely textured performance, capturing the essential decency of a guy who’s often kind of a jerk. Lloyd refuses to yield to his father, who hopes for reconciliation, and you understand exactly why. You also see how clinging to his anger has only worn Lloyd out; surrendering to feelings is a relief.

But it’s not always easy—something Mr. Rogers knew and Hanks channels, in this delicately calibrated performance. At one point Rogers tries to communicate with a distraught Lloyd through his alter-ego, a raggedy puppet known as Daniel Tiger. Lloyd, fuming, practically swats the puppet away; he also asks why Rogers doesn’t get a new Daniel, as this one has had much of its fur loved off. Rogers shifts the conversation and somehow gets Lloyd to admit that he, too, had a childhood friend, a stuffed creature named Old Rabbit, to whom he could confide all his secrets.

That looks terrible on paper, but in the hands of these actors, you buy it. And when Rogers makes a seemingly offhand remark to Lloyd—“Everything mentionable is manageable”—it’s as if an incredibly complicated lock has been sprung with an astonishingly simple key. This is a movie that’s both entertainment and spiritual toolkit—take from it what you need.

Some of what happens in A Beautiful Day is tinged with fantasy and some events are viscerally real. Does Mr. Rogers really show up at Lloyd’s father’s house with both a pie and just the right comforting words? Or is that just a thing Mr. Rogers, in our dream of him, would do? Does it matter? At one point, Lloyd and Rogers ride a New York City subway train and their fellow passengers—a mix of black, Latino and white kids, some grown-ups, a few cops—recognize Mr. Rogers and, almost subconsciously at first but with escalating gusto, serenade him with the theme song to his show. The sequence has a dreamy feel; it couldn’t possibly have happened. But Junod, the character for whom Lloyd is a stand-in, saw it himself; it’s there in his Esquire piece. We humans are a sorry lot, prone to all kinds of self-deception, which often only leads us to lie to others. But we still know how to sing, even in a place that some days seems god-awful, the New York City subway. There’s hope for us yet, and it lies in making the most of this beautiful day.

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