The biggest reason why many people choose Mediation is privacy. Any discussions are protected by a confidentiality agreement, and you avoid airing the details of your conflict in the public forum of the courts. Mediation is a safe place to talk about whatever you need to talk about, without worrying about judgment or embarrassment.
Mediation is also typically much faster and less expensive than the traditional, adversarial approach. An agreement can be reached in weeks, rather than months. Mediation fees are typically lower than an attorney's, and fewer hours are involved in the process. Those parties who choose to also consult an attorney are better prepared, with most issues already resolved, and so need much less time with their attorney.