A lot of mediators will start off a mediation with the disclaimer that, unlike a judge, they have no power to decide the case. They are merely there to help facilitate a resolution. This strikes me as an unhelpful way to begin the proceedings. It seems better to remind the parties of all of the benefits of mediation that they cannot obtain in court. For example, mediation can provide an opportunity to make arguments directly to the opposing parties, as opposed to making arguments to a neutral party like a judge. Mediation also empowers the parties to craft their own solution to the dispute, rather than placing them at the mercy of a judge. Mediation makes parties the masters of their own fate.
As to the mediator’s powers, I like to remind parties that a mediator has some amazing powers that judges only wish they had. For example, a mediator is allowed to conduct ex parte communications with the parties, which is generally forbidden in court and in arbitration. A mediator is also allowed to keep information confidential from the other party. A mediator is allowed to consider information that a judge could never consider because it is not relevant or admissible evidence. Anything that the parties and the mediator consider important can be discussed in mediation. By telling the parties that we can do things in mediation that can never be dreamt of in court, we start the mediation off on a positive note, and make the parties more receptive to making the most out of the process.
(illustration from RIPL research group at Arizona State University-they seem to be studying math, not the kind of mediation I know anything about)