Is there any difference between the following two expressions?
I have a problem with somebody/something.
I have an issue with somebody/something.
Of course, I want to make sure that these two are correct.
Thank you very much.
As the Administrator said they're both correct.
To my knowledge, referring to an "issue" rather than a "problem" has emerged from the thinking that, in modern everyday life, especially on a professional background, you should generally avoid to call something a "problem", because it may indicate that you are really at a point where you are unable (or even unwilling?) to proceed with your intended actions. So, calling something an "issue" instead of a "problem" is maybe a form of euphemism. "Issue" doesn't sound as "problematic" as "problem" to most people, as far as I know.
My personal opinion is that you should name things as they are, and I'm not promoting euphemism, so personally I'd agree to say that both versions are the same. But I suppose that saying "I have an issue with somebody" is a more polite or more careful statement than to say "I have a problem with somebody". A "problem" is, well, a problem, but an "issue" can be something less obstructive or something less visible and more subtle.
Note my small correction in your first paragraph.
In your second paragraph, I think you intend to say that you have the same opinion as others. If so, it is better to say:
I('d) agree (with Tdol) that both versions ..., or possibly: I('d) agree (with Tdol) in saying that both versions ...
If you agree to do something, you say that you will do it.
(I'm not an English teacher, and my business experience is virtually all software development.)
I agree with everyone who said that the idioms are fine.
But, if you're talking about something like a technical meeting, I wouldn't say "I have a problem with John." That's very, very strong. I'd be much more likely to say something like "I have an issue with the way John changed the procedure." (I think the idiom is fine; it's just that I think it's much stronger than people have said.)
Plain "I have a problem with John" is very agressive. It's the sort of thing you'd say when things have gotten really bad. Even "I have an issue with John" is very emphatic. I think people are much more likely talk about something John did, rather than talk about John himself. "I have an issue with the way John changed the process", rather than "I have an issue with John. I don't think the way he changed the process is the right thing to do."