Since accurate productions of Brecht-Weill's play are a lot rarer in the early 21st Century, it's likely that most viewers are more aware of the production via Pop-Cultural Osmosis from Alan Moore's comics and the many popular music covers than from the actual play.
"What Keeps Mankind Alive" was covered by Tom Waits, with accurate lyrics.
Darkness-Induced Audience Apathy: Any accurate version of Brecht's play is going to result in a story of Evil Versus Evil, where the law (represented by Tiger Brown) is ineffectual, hypocritical and compromised, where women are consigned to endure Domestic Abuse, and a totally irredemable asshole becomes a Karma Houdini, rewarded for all his murders, rapes and abuse, and beggars continue to be under the thumb of his father-in-law with no real sense of moral authority anywhere.
Draco in Leather Pants: Macheath is the one character most audiences seem to like best, even though he is intended as a despicable villain. Though of course most productions water down the violence and nastiness of the original a lot, if portrayed the way Brecht wrote him - an amoral pimp, human trafficker and child murderer - this trope is less in play.
Ensemble Darkhorse: Jenny Towler is a minor character and in the original production, her famous song was actually sung by Carola Neher's Polly. But thanks to Lotte Lenya's stage presence and better vocal range, Pirate Jenny became her song, and as such a throwaway minor character became an iconic feature of popular culture, inspiring Bob Dylan and Nina Simone in particular.
My Real Daddy: On account of what some consider Brecht's unique approach to collaborationnote The most uncharitable view, the one most hostile to Brecht's politics and dramaturgy, see it as merely taking credit for other people's work and pretending a collage and group production was really his vision, a lot of critics and scholars have wondered about the authorship of this play. While the overall vision, theme and commentary is by Brecht, other parts of the play come from his collaborators:
Some note that a surprising amount of The Threepenny Opera is already there in John Gay's original Beggar's Opera (Macheath, Tiger Brown, Polly, Lucy, Jonathan Peachum, the Deus ex Machina ending), the Love Triangle plot. The main additions are largely modernized sex and violence, an anti-imperialist subtext and pitch-Black Comedy (plus Adaptational Villainy for all the characters) replacing the lighter farce of the original.
A German production of The Beggar's Opera preceded Brecht-Weill's version, and Elizabeth Hauptmann (Brecht's collaboratornote and also one of his girlfriends) translated the English original into German, and even contributed a few lines to the lyrics, which were indeed written largely by Brecht. As such some have argued that Hauptmann be co-credited with Brecht-Weill for her work on the play.
Others argue that the most important thing about the play is really the music and that Kurt Weill was the real genius behind the production. Since the music went to produce many Breakaway Pop Hit and became immensely influential on popular music for the rest of the century, it's a fairly strong argument. They also note how much of the production was made up as it went along, since "Mack the Knife" was written near the end because the original actor of Mackie (Harald Paulsen) complained how little screentime he had and wanted a song centered on him, Brecht-Weill complied but gave the song to the street-singer, a new character invented solely to sing the Moritat, rather than give Mackie any more stagetime. Likewise, many credit Lotte Lenya's excellent stage presence for significantly expanding the character Jenny. Brecht and Will might have written the song, but it was Lenya who made it Pirate Jenny.
Older Than They Think: Several of Macheath's songs are taken from the French renaissance poet (and professional criminal) Francois Villon. It works in the other direction, too: A lot of people are more likely to connect the "Moritat" with Bobby Darrin or Louis Armstrong than Bertolt Brecht.