Sometimes, students write sentences like: "I was in London last year and I could see the Tower of London." This is wrong. The correct options are: "I was in London last year and I was able to see / I managed to see the Tower of London." The correct options communicate the idea that something was possible in the past, and you did that possible thing in the past. "could" does not express this. However, (a little confusingly) "couldn't" is an option for negative situations.

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)

The contrast is clear in the sample sentence which you have given us: "The police were able to find out that he could speak French" It is correct to use 'could' in the phrase 'He could speak French' because this refers to a constant fact about this person. His ability to speak French was not restricted to one occasion. However, you have to use 'were able to' or 'managed to' in the first clause of the sentence. You cannot say 'The police could find out..' , because this refers to one single occasion. The police did some investigations and found out that that the man was a French speaker - this is a specific thing that they managed to do on one particular occasion. It is not a general ability, so 'could' is not possible.

I used to do martial arts for about 10 years. In those days, I was able to hit a target so fast, but I cannot do so now. :(