In addition to what Steve White says: if you're hearing these words in songs, "issue" can be more dismissive, more socially indifferent, less caring, to the person who has the problem. Its a slight connotation. But if you say "that guy is having some problems" it can mean that maybe you sympathize with him a little and would not wish him ill and maybe you would think it good if his problems went away. But if you say "that guy has issues" its like his problems are either superficial or his own fault especially because of his personality, or that he has problems with others because of his personality, something like that. The word "issue" used in business, is, as Steve White says, "problem" but has no such social connotation as I just described here. Its a softening of the word "problem" like its just an every day thing that comes up and needs to be handled. Also a joke for you. Since the word "issue" is also used with magazines and things that one subscribes to, like 我订阅一份画报, there is a joke on the double entendre "That guy doesn't have issues, he has a whole subscription." in other words, his socially dismiss-able problems are so numerous that he has a "subscription" as if to a magazine.

Is this post meant for correction? Or is it a question? I'll assume it's a question. :) "Problem" just means some difficulty or puzzle or suspense that requires solution, or resolution, or being fixed. The word "issue" has more than one meaning, but one of its meanings is as a fancy synonym for "problem". In business, people often prefer to use the word "issue" instead of "problem", because "issue" sounds like less of a big deal. So it's used euphemistically. But when used in that way, "issue" means exactly the same as "problem". :)