They're both beloved character actors who have appeared on notable sci-fi shows, but no, Anthony Rapp is definitely not Alan Tudyk. It's not an uncommon phenomenon for audiences to get actors confused due to similar appearances. Mad Max: Fury Road star Tom Hardy gets confused with Upgrade actor Logan Marshall-Green quite a bit, and people often mistake Henry Cavill with Matt Bomer.
Anthony Rapp first made an impression by playing Mark Cohen in acclaimed musical Rent, and later starred in the 2015 film adaptation. Rapp also plays Lieutenant Commander Paul Stamets on CBS series Star Trek: Discovery. The actor has a rich CV and has jumped back and forth between theatre, movies, and television throughout his career, but due to his resemblance to Alan Tudyk, some viewers can get the two actors confused.
Tudyk once directly addressed this himself during a Wired interview, admitting people often believe he was in Rent due to the resemblance between him and Rapp. Tudyk also has an extensive CV and is probably best known for his role as pilot "Wash" in Joss Whedon's sci-fi series Firefly and subsequent movie spinoff Serenity. He also played sardonic android K-2SO in Rogue One: A Star Ways Story and is set to reprise the part for Disney+ live-action series focusing on Diego Luna's Cassian Andor.
While Anthony Rapp and Alan Tudyk definitely bear something of a resemblance to one another, particularly in photographs, in motion it's not that pronounced. Star Trek: Discovery has brought Rapp a new level of fandom, though he doesn't have much in the way of other sci-fi projects in his prior work. Tudyk has been much more prolific in this regards, having appeared in I, Robot, Transformers: Dark of the Moon, and most recently DC show Doom Patrol as Mr. Nobody.
Tudyk has also loaned his voice to video games like Halo 3, Injustice: Gods Among Us, and Call Of Duty: Advanced Warfare. Anthony Rapp has a rich history with the theatre, and aside from Rent, he's starred in Little Shop Of Horrors, Henry V, and Hedwig and the Angry Inch. The two haven't appeared in a movie or TV show together to date, though Tudyk still likes to poke fun about their resemblance to each other on social media.
The Anthony Rapp / Alan Tudyk confusion is likely to continue for future projects, but given the quality of work both actors produce, the comparisons aren't exactly an insult either.