Hop hop! Hop hop!

Terry Teachout has it right on Alban Berg's Wozzeck:

As for the opera itself, regular readers know I’m no fan of Austro-German expressionism, but Wozzeck is one of the supreme masterpieces of twentieth-century opera, a work so overwhelmingly compelling as to overwhelm any possible objections.

Wozzeck, a 20th-century opera based on the weirdest play of the 19th century (a play that, though written in 1837, reads like a work of the 20th-century avant-garde), is often considered the atonal opera for people who don't like atonal music. In fact, Berg is generally the most accessible of non-tonal composers. Even after Wozzeck, when he adopted Schoenberg's abtruse twelve-tone system, his music remained accessible; his violin concerto is as close to a popular concert staple as a twelve-tone composition can get, and his last opera, Lulu, is gorgeous-sounding despite its ugly subject-matter.

As to why Berg's music is easier to get into than most atonal music, I think it has to do with his understanding of voices -- instrumental voices and human voices. Many twentieth-century composers were cursed with an inability to write for voices and instruments in a manner that would sound attractive when performed. Schoenberg was often like that in his vocal writing, coming up with bizarre vocal lines that made a singer's voice sound unattractive. That, not the twelve-tone system itself, is what makes a lot of his music sound ugly: he had worked out the system, but he hadn't fully figured out what the music would sound like. Berg wrote for voices in a very challenging and difficult way; roles like Marie in Wozzeck and Lulu can kill a soprano's voice. But while music is hard, if the singer is up to the challenges, the music will sound great: Berg had a knowledge of the human voice that allowed him to write music that sounds beautiful when sung, just like Mozart or Verdi or either Strauss. His violin concerto is similarly good at making sure the music sounds right and appropriate for the violin.

A lot of the twelve-tone music that came after Berg (who died young) tended to be written by academics, like Milton Babbitt, who didn't have a lot of practical experience with writing for performers, and often wrote music that sounded better in theory than in practice. If twelve-tone music never became popular -- and it never did -- it may have more to do with that than any inherent problems of the twelve-tone system. It could be that if Berg had lived he'd have come up with the first genuine twelve-tone popular favorite; he could have done it if anyone could have.