One of the most important distinctions in Plato is that between dialectic and eristic, two forms of arguing. Roughly, I would define them this way:
Eristic is a competitive disputation in which opponents battle to defeat their disputants. Each side is convinced of the rightness of its own opinion and of the other side’s error. Notice that eristic always involves the assertion of mere opinion as the truth. Typically the dominant emotion is anger, and each side will argue using the most uncharitable construction of what the opponent is saying. Both sides suspect the other of bad motives and reject any argument which reaches a conclusion different from the claim they are defending. Therefore, neither side opens themselves up to refutation. The disagreement offends both parties, and the only form of agreement that is acceptable is the surrender of the other. The eristic mindset is governed by a simple heuristic: a friend is one who agrees with me, an enemy one who disagrees.
Dialectic is a cooperative search for the truth beyond opinion. Lack of agreement is a welcome opportunity to test one’s own opinions/beliefs for error. Common agreement is sought and it doesn’t matter to either party which initial opinion is closer to the agreed-upon conclusion. Anger is not a characteristic of dialectic. Thoughtful consideration of the other’s beliefs and arguments, even when they would put one’s own at risk, is the chief characteristic. Each side is willing to state their true beliefs and expose them to the hazard of refutation. Friendship is no longer contingent on agreement/disagreement but is based on a mutual willingness to help the other and be helped by him/her.
What makes eristic frustrating is that the attempts to persuade the other often seem so counterproductive. Eristic is essentially scandalizing, a scandal being a situation in which attempts to remove an obstacle makes it more entangling. Opposing an eristic opponent increases, rather than decreases, the strength of their opposition. The spirited/competitive part of the soul, the thumos, is excited by opposition and becomes all the more determined to win supremacy.
The chief discriminator between the two is the type of object that each one pursues:
You should not make such wholesale charges against the majority, for they’ll no doubt come to a different opinion, if instead of indulging your love of victory (philonikia) at their expense, you soothe them and try to remove their slanderous prejudice against the love of learning (philomathia), by pointing out what you mean by a philosopher and by defining the philosophic nature and way of life, as we did now, so they’ll realize that you don’t mean the same people as they do. And once they see it your way, even you will say they’ll have a different opinion from the one you attributed to them and will answer differently. Or do you think that anyone who is gentle and without malice is harsh with someone who is neither irritable nor malicious?
…
[The] harshness the majority exhibit towards philosophy is caused by these outsiders who don’t belong and who’ve burst in like a band of revellers, always abusing one another, indulging their love of quarrels, and arguing about human beings in a way that is wholly inappropriate to philosophy… Republic, Grube/Reeve translation, 499d-500b
Both polemical disputants and dialectical participants are passionate, but oriented toward vastly different objects of desire. Victories cannot be shared without diminishment, whereas learning can. Philonikia (love of victory) is essentially scandalous, whereas philomathia (love of learning) is not.
This passage also contains one of the motives for Socratic irony. The distinguishing feature of Socratic rhetoric is its insistence on questioning and on its ironic detachment from answers. Socratic irony is a stance of surrendering the rivalrous philonikia, the desire for victory, in those win/lose situations that excite the thumos. In a competitive struggle for such goods — particularly for metaphysical objects like honors, power, popularity, fame — someone must be sacrificed/defeated to remove the scandal. Socratic irony is an act of removing the acquisitive mimetic postures that lead to mimetic rivalry and replacing them with the kenotic posture of ignorance and preemptive defeat. In a world of competitive polemical debate, to admit ignorance is to be defeated. To give up claim to the trophy or victory at the level of eristic clears the way for a common pursuit of a sharable good such as learning.
shared ignorance 
Socrates isn’t always effective at tamping down the thumos of his interlocutors, is he? Some may find his ironic detachment more, rather than less, maddening; though that is how we can tell the true students from the false. What do you make of that?
Not always. There is no infallible method for dealing with another’s recalcitrance. Some will never give up the notion of argument as a win/lose affair, even if you do cede them victory. I almost regret the phrase “ironic detachment,” since Socrates is never anything but passionate, but there is a detachment from the trophy-motive that drives polemical exchanges in order to attach all the more passionately to the fruit won through dialectical engagement.
Those who get most angry at Socratic irony are those who interpret it as a sophisticated ruse (like Thrasymachus, Callicles), but I am not sure it is all just pretense in Socrates’ case. Interestingly, the eristic types are not offended by the ironic flattery of themselves — which is the false aspect of Socrates irony — but they are offended in Socrates’ claim that he doesn’t know. That latter claim is not disingenuous, I think.