Most theater aficionados know the storied history of the musical “Rent.” Composer Jonathan Larson drops dead of an aortic dissection on the eve of the opening in January 1996. Heroic cast soldiers on, despite their grief. Show opens to rave reviews and moves to Broadway. With its themes of multiculturalism, homophobia, and AIDS, the musical based on “La Bohème” becomes a cultural phenomenon among Generation X. Groupies known as “Rentheads” camp out daily at the theater to score golden $20 tickets for seats in the front row.

That back story has become inseparable from the show itself and contributed, at least in part, to its wild success. But what many might not know is that Anthony Rapp, the fresh-scrubbed actor who created the role of videographer Mark Cohen, was undergoing a family tragedy at the time of the show’s meteoric rise. His mother was dying of cancer, and he shuttled back and forth from New York to Illinois to be at her bedside.

Rapp intertwines both tales in his one-man show “Without You,” which is based on his 2006 memoir of the same name. The show is making its Boston premiere at the Modern Theatre through Sunday, before moving to the Edinburgh Festival Fringe and London’s Menier Chocolate Factory.

With his chiseled chin and strawberry-blond hair, Rapp is still as engaging a performer as he was when he landed the role in “Rent” while working a “survival job” at Starbucks for $7 an hour. His portrayal of his mother, with her flat Midwestern accent and gentle demeanor, is lovingly drawn. She calls him ’Tonio. He calls her Mama.

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Performing on a spare stage with a five-piece band, two chairs, and a baby book, Rapp gives new resonance to standards from “Rent,” particularly the funereal celebration “Seasons of Love.” He also penned new songs for the piece, which are largely unremarkable, with the exception of “Wild Bill,” a country-tinged tune about his mother’s nickname for her disease.

In its own way, “Without You” shares many of the flaws and strengths of “Rent.” It’s heartfelt, youthful, poignant. In his Pumas and plaid shirt, Rapp is earnest and charming. But while the piece is sentimental by design, it fails to pierce an emotional vein. The death of one’s mother is profound at any age, but I never caught myself fumbling for a tissue or raising a hand to wipe my eyes.

Part of the problem is that there’s too much Larson and too little Rapp. There’s no tension in the story of the composer’s death, because we all know how that one ends. And there is surprisingly little tension in the way Rapp recalls his mother’s battle with cancer. He says he needs to resolve her reservations about his homosexuality, but he barely registers anger or regret. He leaves out a story he included in the memoir about a back-alley brawl he had with his lover during this period, a physical manifestation of his rage against the dark. One gets the feeling that his endearing urge to respect and honor his mother has nudged him to pull a dramatic punch.

Director Steven Maler’s production aims to ratchet up the drama by shooting harsh red and white spotlights out into the audience, but the effect is merely alienating, not wrenching. And in the intimate space of the Modern, the band sometimes drowns out Rapp’s voice. The quiet moments are anything but.

For the most part, the storytelling is linear and literal. The piece does rise above the prosaic when Rapp recalls a story about his mother, a nurse who once cared for a terminally ill boy who seemed afraid to let go. She tells the boy’s parents they have to give him permission to die. That single image of Rapp cradling an imaginary child evokes the visceral lose-your-breath feeling that the rest of the piece strives so valiantly to create.

But the bespectacled and boyish Rapp commands the stage, and with some tweaks, his solo show could become less married to “Rent” and more transcendent. There is certainly a built-in audience, with all those Rentheads who are not so young anymore and who have surely experienced losses of their own. They all rose to their feet at Wednesday’s packed performance, a sign that this tale of loss may have a life of its own.