The Saxon textbooks and the AoPS materials approach math differently. For both, the goal is the same: understanding of mathematical concepts, and mastery of problem-solving procedures.
However, the programs reach this goal via different paths. AoPS is primarily conceptual in its approach. Conceptual math is shorthand for mathematics instruction that focuses on discovering the reasons why operations work the way they do. Saxon tends towards the procedural method; it often begins to teach students new concepts through laying out and practicing a series of steps (procedures).
Obviously, all students need both concepts and procedures. However, students who have a strong mathematical inclination (and so are able to grasp abstract concepts a little more easily) will probably flourish in the AoPS classes, while those who benefit from more support and step-by-step instruction should consider the Saxon classes
In addition, the AoPS classes are somewhat more intensive, meeting 3 contact hours per week rather than 2, giving students an additional class period to explore and master advanced concepts. There is also an added emphasis on out of class work. This additional instruction and grading time accounts for the higher tuition rate for AoPS.
Both classes will provide your students with strong mathematical skills.