Conversations on internet forums are, in my opinion, more difficult because you have varied, unknown elements who can become very confrontational once you state an opinion. They have the anonymity to be as harsh or hateful as they like, whereas in person they might use more discretion and tact.
My best advise for a difficult conversation is to think about what you want to say or what needs to be said beforehand so that you don't lose your cool or your train of thought in the heat of the moment. If you're mad about a situation, of course, it's best to wait until you've cooled down to talk to someone so you don't overwhelm them with your emotion.
In trivialities, I will agree that "who's right and who's wrong" doesn't matter. But in larger issues, it definitely does matter and it all depends on how much you care that it also matter to the person you're conversing with. You can skip around the issue a little to determine how they are most likely to be convinced of your argument, if at all.
In person, I have no time to think about what the other person is saying and respond based on that consideration. I can prepare what I'm going to say, but if the other person has a perspective that I hadn't considered, then I don't have time for careful consideration. I can try to "win" by overcoming his perspective with the strength of my ideas, but that solution removes the possibility that we'll find a better answer that incorporates both of our perspectives. I can try to listen and consider his perspective during the conversation, but in doing so, I risk letting him "win" by overcoming my position while I think about what he's said.
The beauty of some form of written discussion is that I can carefully consider every point in context. I can let whatever emotion arises within me play itself out before trying to respond. I don't have to spend energy fighting the emotion. I can let the emotion blow through my mind like a storm and then consider things when I find the calm on the other side. I can read the other person's words over and over in order to understand whether his positions are internally consistent. If the discussion leads us to a better idea, I can find that idea more easily. If the discussion only proves me right, then I can explain more carefully exactly why I'm right.
Of course, the other side of this coin is that much of communication is not only in the words. In writing, we lose body language and voice tone completely. If both people in the discussion have some skill in writing what they want to say, then this loss can be overcome. If one or both of them cannot communicate well using only the words, then the loss is nearly impossible to overcome.