Contract disputes are traditionally resolved by judges or juries. Besides the costs involved, this entails putting resolution of your dispute in the hands of a judge or jury who does not know the facts or circumstances anywhere near as well as you, and may not, because of legal rulings, even get the complete facts or circumstances to make their decision. Further, it is unlikely that your judge will be well-versed in the practical aspects of construction projects or standards, and more likely than not that anyone that could serve on a jury with such knowledge will get "struck" from the jury by one side or the other and so not be able to serve on the jury.
Alternative dispute resolution, and in particular mediation, provides a mechanism to avoid this, and have your dispute considered and resolved with construction subject matter experts, either as neutral facilitators or arbiters. Such methods can range from very detailed processes with extensive pre-resolution "discovery" to much more informal discussion sessions. They can be just between project personnel to try and resolve change orders, or involve counsel for more complex or value related disputes. The beauty of ADR is that the parties get to shape its form and process. They can mirror trials but with subject matter experts, or be conducted more summarily with rules of evidence or procedure tabled.
In the end, though, your dispute is resolved or facilitated by means of your own choosing, and generally more quickly and in a less costly fashion. So why isn't it used more often? That's a complicated question but the answer generally lies in either one side or the other's perception of leverage, or their counsel's unfamiliarity with such process, and preference for the "old ways" of resolution. To me, the more important question about ADR is not whether to do it (you should) but is rather when you should do it. This hinges primarily upon whether there is sufficient information already known or exchanged to make an ADR session effective. Typically, the sooner that is the better, because resolution is more likely since positions haven't yet been entrenched, and quicker resolution is usually less costly.