Let me give you my two cents input: Not to disagree with anyone but at least within the circle of people I work with on a daily basis; an issue has a softer connotation and when not dealt with it properly it could lead to the creation of a problem. Ex. John and Peter don't agree on the rate they should use; if they don't deal with this issue now, it might become a problem later.
Pescara is right to say that even when you think you have a problem, is sometimes better to say: let's work on this issue or let's try to resolve this issue than to use the word problem. A psychologist will never say: Let's work on your problem, but let's work on this issue. Is he or she being ambiguous? No, it's being correct and professional.
Serif quoted the dictionary correctly to say: An issue is not yet a problem but a point of question or a matter that is in dispute. Therefore, I don't find it annoying when people try to be polite and use this euphemism (although it might not be the case) instead of being aggressive. It may be that they are simply using the word issue correctly as indicated in the dictionary, common sense.
As explained here, I'd have no problem

in using the word "issue" this way because it corresponds with the original meaning: a subject of discussion or controversy. But I'd have more difficulty in saying
John and Peter have an issue about the rate they should use. On the other hand I might easily say
The issue (the subject of the discussion/dispute) is the rate they should use.
In my (old) vocabulary an unresolved
issue can become a
problem, but I wouldn't call a
small problem an
issue, I don't see it as a synonym.
If a machine breaks down, I say we have a
problem (something that creates difficulty). If we disagree about whether to repair it or replace it, that is an
issue: how to solve the
problem.
To summarize: if a bridge is broken and I can't cross the river, that's a problem; in no way would I call it an issue, not because of the
size of the problem but because it's something that exists objectively, not a matter of opinion.
If we discuss why the bridge is broken, why it hasn't been repaired or why a boat hasn't been provided, that's an issue, not a problem.
The discussion of this issue

should probably be transferred to the English Only forum.