A "problem" is one thing. It is like a puzzle. It can be "solved." "My problem is that I need to get to Malden and it's a long time until the next train. Wait, let me check the schedule--I can take the bus and the subway instead. Problem solved." "Trouble" is just an overall bad situation. "My problem is that I need to get to Malden and the last bus is gone and I have no money. I'm really in trouble now." A "problem" isn't necessarily bad. It can be a task to be done. "My physic teacher gave me twelve problems to solve." "Trouble" is always bad and always serious. The forty-year war-like conflict in Northern Ireland was called "the troubles."

The first thing the spring to mind is that "problem" is countable and "trouble" is uncountable. Also, a problem is solvable.

I believe trouble can be used as a verb: "I don't want to trouble you" "what is troubling you?" As a noun, it means 'difficulty' "I have a lot of trouble with math"; 'problems' "we're having some trouble here" I'm not sure, though, I hope someone has a better answer for you, my friend!