There is not much difference. "Difficulty" merely suggests something blocking your way. It puts the stress on you. You are stuck. You are blocked. You can't move. You may not even understand why not. "Problem" puts the emphasis on the thing that is blocking you. In order for it to be a "problem," you have to be able to see what the problem is, even if you can't solve it. "Problem" suggests a situation where you can think logically and analyze possible solutions. Often, you could use either word, but in some situations one or the other would express the situation more clearly. For example, "I am having difficulty with this problem." Or, "I am having financial difficulties; I don't know what to do." "Your problem is that you didn't make out a household budget and stick to it." "Learning a language presents many difficulties. Expecting a language to be logical can be a difficulty in itself. Accept the irregularities as part of the language without treating them as problems that can be solved with logical rules."

In may cases they are same problems - origin of the word is riddle or question. The emotion behind the issue thens to be less or neutral difficulties - implied that skill or ability can be used to overcome. The issue can be more emotional.

If you have a problem it means that you need to solve a difficult situation, or you have an obstacle, the term can be used also in technical/scientific/mathematical fields. Having "difficulties", this is a synonym in some way and it's used in the interpersonal/social context, it refers to human beings, life experience, work situation. Somebody who is going through a diffucult period is "in difficulties" or "fell into disgrace" because he/she is "broke"...no more money to live.

Problem is more like an obstacle, something to overcome. Difficulty is more like a challenge, something that makes it harder to perform a task.