What is the difference between these words?
As a first and very approximative answer, the following can be said:
sorun = question (sorun ne? = what is the question?)
problem = problem (problem ne? = what is the problem?)
mesele = issue (mesele ne? = what is the issue?)
dert / sıkıntı = trouble / nuisance
All the words you have asked about have an extremely wide range of uses both in literary and colloquial Turkish.
What matters is really the context. For example, "Derdin ne?" can both mean "What is your trouble?" and "What is the matter with you?"
A relatively recent use of sıkıntı is "Sıkıntı yok", which means "No problem", "It's okay", "All right", etc.
I hope the foregoing discussion helps at least as a starting point.