No two private schools are the same, just as no two public schools the same. There are so many studies whether private or public schools are better that almost anybody can find one that supports their opinion.
However, there are certain things common for most private schools that everybody agrees on.
Private schools tend to have smaller class sizes, and smaller student to teacher ratio. Therefore, students are able to get the attention they need. The classrooms are more relaxed, students get to work freely with some assistance from teachers, allowing them to learn how to solve problems and work at a more individualized pace while still achieving academic goals. These schools are independently funded, responsible for their own budget, and therefore, able to fund more extracurricular activities/programs, attract better teachers, and develop a more advanced curriculum. In many private schools, students stay with the same teacher for a couple of years, which helps the teacher to know their students better and aids in addressing their individual needs much better.
As parents who care for the education and the future of our children, we cannot be blind to the reality that education is not at the top of state of Idaho priorities. In fact, Idaho was ranked 38th among the states in 2009-2010 (latest statistic available) for percent of federal and state expenses dedicated to school. This number when translated in absolute values gets even worse leading to a 47th position for dollars spent per student enrolled (fall enrollment 2010-2011).
Source and detailed data is reported in the following table.