The word problem is connected with the word "solution." There is usually some specific way to resolve a problem. If there's a problem with your computer, the computer repairman can fix it. If you have a problem in your relationship, you can talk to the other person to find a solution.
The word trouble is more connected with general difficulties. We often use the phrase "having trouble" to mean "having difficulty":
I'm having trouble sleeping. = I'm having difficulty sleeping.
If someone "is in trouble" or "gets in trouble," it means they are in a bad situation:
The boy got in trouble for stealing a toy from the store. or ( he will be disciplined)
If sales don't start improving soon, our company will be in trouble. or ( the company will have difficulty surviving)
The word problem is countable, and the word trouble is usually uncountable. We can say "a problem," but not "a trouble."
There's a problem with my online bank account.
I'm having some trouble logging into my bank account.
The speaker talked about five major problems in society. (notfive troubles)
Strangely enough Venus completes an entire orbit around the sun before it manages to turn on its axis once. This means that its day is actually longer than its year and in Venusian time, World War II ended only 56 days ago.