Does "he has some issues to work out" have some special meaning?
Is it an idiom? How is its meaning different from "he has some problems to solve"?
There's a particular implication of the "issues" or "problems" involved being mental or emotional ones. It's more closely equivalent to "he's a little neurotic" than "he has some problems to solve".
Yes it is an idiom. Polite way to say he is unstable and needs to see a psychiatrist or at least get a grip :)
Yes, "he has issues" would imply mental or emotional instability. Extending "instability" from an individual's mental state to the nation's financial state, a business publication referred to the recent wave of American bank failures as "American issues." But perhaps that's such a bad usage that it doesn't deserve notice.
In this context the "issues" are usually psychological. It's elided out of politeness. "Problems to solve" could just as well refer to homework.