I recently finished the mediation of a complex federal court action where the parties started off over 20 times apart -- the plaintiff's demand was more than 20 times the defendant's offer. 
The case settled in part because the directed conversation during the mediation revealed to the respective attorneys that they were too close to their case to objectively value it.
Now, what do I mean by that -- don't we WANT lawyers to be intimate with their cases? Of course we do -- especially clients. But frequently lawyers just know too much about the case, all the subtleties of the law, all the twists of the facts, the hidden gems in the testimony.
As a result, it becomes very easy for them to discount even "bombshell" revelations from the opponent. They will often see all the individual "trees" of their law and evidence as providing great strength and cover from the arrows of the opposition.
The problem is that they overlook that the fact-finder -- be it judge or jury -- will never know their case with the intimacy that they know it, will not live with the case, day, night, and in their dreams, the way litigators live with a case as it rides the rails towards trial. As a result, the fact finder tends to either focus on a few of the most obvious elements of the case (obvious to them, at least), or to see a monochrome, a monolith, with few shades, where the attorneys see every possible shade of green in their forest.
It is this very tendency which lays behind the increasing use of mock jurors and trials as part of the litigation preparation -- even in small or solo practices. More often than not, attorneys are surprised by what they learn from the mock experience. Mediation can have the same impact.
Often, mediation is the first time that the lawyers and their clients will hear the weaknesses of their cases ably presented by the other side. This will come at several points -- at the outset with the opening statements, again during the issue-identification phase (when we often learn that the two sides' opinion of what is significant may be quite different), and in the negotiation phase, when perhaps facts or testimony not previously considered significant suddenly is made to fit within the case theory of the other side.
The mediator actively assists in the new evaluation of the case by asking probing questions, at appropriate times, for example, regarding the impacts of factual determinations on strategy and anticipated outcome, or about potential juror reaction to testimony when it is cast in new lights. It is the rare mediation in which both sides do not emerge with some new take on their case. Even where the mediation does not resolve the dispute, for the skilled counsel the mediation experience can become an important part of shaping the narratives to be presented at the trial. (That, alone, should make every litigator interested in pursuing mediation at some point prior to trial.)
One of the reasons mediation works is that it can help counterbalance the unintended consequence of lawyers knowing their case so well -- overconfidence.