They are very similar in concept and use. Both express the concept of difficulty. However a problem is "something" that is causing a difficulty. So in this sentence, I am having trouble starting my car, the problem is that I have no petrol. So trouble is general, problem is specific.

Trouble and Problem are indeed very similar, they are actually two ways to say the same thing. When something is in trouble, it implies that there is a problem. And also when there is a problem, there will likely be something that is in a troubled state. Trouble is a state of difficulty. In other words, trouble is the "state of having a problem". For example, "My car is having trouble starting." and "I have a problem, my car is not starting." are two ways to say the same thing. As said in the previous answer, trouble is usually about something general (meaning that you don't know where the problem is specifically), and when you know the problem, only then would you refer to it directly as a problem.

trouble is the result of the problem. while problem is the result of wrong doings/choice.