Problem
Part of speech: noun
Definition: A difficulty that has to be resolved or dealt with. A question to be answered, schoolwork exercise. A puzzling circumstance.
Example sentence: The whole problem with the world is that fools and fanatics are always so certain of themselves, and wiser people so full of doubts.
Trouble
Part of speech: noun
Definition: A distressful or dangerous situation.A difficulty contributing to such a situationEfforts taken.A malfunction, as in "heart trouble".A violent occurrence.
Part of speech: verb
Definition: To bother; to annoy.
Example sentence: A 'No' uttered from the deepest conviction is better than a 'Yes' merely uttered to please, or worse, to avoid trouble.