The similarities seem to be that each is an intricately plotted courtroom drama, and each features a scandalous woman accused of murdering a millionaire.
Here's a plot summary of Mary Dugan:
A Follies beauty, Mary Dugan, is on trial for the murder of millionaire Edgar Rice. Her defense seems to be faltering until her young brother, Jimmy, a lawyer just beginning his career, insists he himself replace her present lawyer, Edward West. Mary admits to affairs with several men but wins favor with the jury when she reveals that the money she received was used to provide Jimmy's education. After it is determined that Rice was stabbed to death by a powerful, left‐handed man, Jimmy provides the reason for her initially weak defense. Rice, it seems, was murdered by attorney West, who loved Mrs. Rice, so decided to frame Mary. Time remarked that the play “moves more swiftly than the law with all its ruthless directness. Its plot has the fascinating features of a front‐page murder story.”Here's a plot summary of January 16th:
Bjorn Faulkner has swindled millions of dollars from investors, by investing cash he didn't have in order to control the gold trade. In the wake of a crash, he is facing bankruptcy despite the injection of money by Mr. John Graham Whitfield, a prominent banker whose daughter, Nancy Lee Faulkner, married Faulkner shortly after the loan. On the night of January 16th, Karen Andre, Bjorn Faulkner's mistress of ten years, and Bjorn are in the penthouse at the top of the Faulkner Building in New York when Faulkner falls to his death. The purpose of the play is to decide if it may have been a suicide - or murder.Rand's play also features a gimmick where an audience jury decides who is found guilty.
You can see the similarities
and also the disparities.