Warning: Wikipedia contains spoilers
Basically, the team of screenwriters took the main suspect of the novel, Robert Tisdall, and his unexpected and at first reluctant supporter, young Erica Burgoyne, and left out all the other characters, including Tey's Inspector Alan Grant and even the murderer (!). In other words, in the process of adaptation for the big screen, the episode revolving around Tisdall as a fugitive from justice was elaborated and blown up into a full-length film whereas the subplots and distractors of Tey's novel -- which make it a whodunnit -- were omitted.
Summary of the plot
Christine Clay, an actress who owes her career to her husband, is severely criticized by the latter for playing around with other men. He is in particular referring to young Robert Tisdall, who is staying with (or at least near) them at their country retreat somewhere on the English coast. One night, the actress smacks her husband's face, who keeps calm except for his eyes, which start twitching uncontrollably. But then he strangles her with the belt from a trenchcoat he has stolen. For some reason, he is never suspected by the police. Just when Christine Clay's body is washed ashore, Robert Tisdall happens to be walking along the cliffs. A belt is found next to the body. He runs away to get help or call the police and is seen doing so by two young female swimmers. He is arrested and becomes the main suspect, also because the actress has left him a large sum in her will. After talking to a hopelessly inadequate solicitor, Tisdall doubts if his innocence will ever be established. In the corridor of the courthouse, while there are many people waiting for and attending different trials, he can make his escape. Soon he is joined by Erica Burgoyne, who gives him a lift in her dilapidated Morris. The two get more and more involved with each other -- not romantically at first -- so that the police think she is his accomplice. All Tisdall wants, however, is prove his innocence by getting hold of the coat (plus belt) that was stolen from him.
During their flight, they briefly stop at Erica's aunt, whose seven year-old daughter Felicity is having a birthday party (playing, among other games, Blind Man's Buff). Finally, in a lodging house frequented by tramps, Tisdall encounters the man he has been looking for: Old Will, a china-mender who turns out to be quite sociable. He is actually wearing Tisdall's coat -- without a belt though -- and readily gives it to Tisdall. What is more, he agrees to help Tisdall and Erica Burgoyne find the murderer and clear Tisdall of all suspicion.
In the pocket of the coat they find a box of matches from the Grand Hotel, which eventually leads them to the real murderer of Christine Clay -- her own husband, the one with the twitching eyes. The nervous tic is the only thing Old Will can remember about the man who gave him the coat without the belt. Eventually he turns out to be working at the Grand Hotel as the blackface drummer in a jazz band. While playing the drums, the murderer has a seizure in front of Tisdall, Erica and Old Will and immediately after regaining consciousness confesses his crime. In the end Robert Tisdall and Erica Burgoyne are united, with Erica's father smiling benevolently.