I hope I understood you correctly: you asked about the difference between COULDN'T and WASN'T/WEREN'T ABLE TO, right? When you want to say that somebody did something in a specific situation, use was/ were be able to or managed to (not could). Here are examples:
Amy was an excellent player. She could beat anybody. (She had the general ability to beat other players)
Amy and Lucy played table tennis last night. Amy played great but Lucy was able to beat her. (Lucy, in this specific event, managed to beat Amy)