A question is just a statement that asks something. Ex. Where are you going? A problem (also called an issue) is usually something that is bad and generally requires a challenge to prevent/get rid of. When it's used to refer to questions on exams/tests, it's neutral and just means 'a challenge'. Any other time, a problem is something negative. Ex. 'We need to get rid of him. He's a problem.' 'This roadblock is a problem.' By the way, your question should be "How do you distinguish between a question and a problem?" or "What is the difference between 'question' and 'problem'?".

问题 Question is to be asked ... 我要问以下；I have a question or two, may I ask you ...? Problem is a difficulty, a trouble, that needs a solution, not an answer