I went to a KC area private school and the main things I noticed are that I feel much more prepared in english/math/writing, and less prepared in physics and chemistry.
The rate of kids who went to college from my class is like 99% or something absurd, but I feel like that has more to do with kids being able to afford college tuition than it does getting scholarships.
If you are in the Blue Valley school district, there really isn't a need to go private other than wanting to say your kids go to a private school. Many Johnson County public schools are just as good, and sometimes better at preparing kids for college. Most kids attend Aquinas/Rockhurst/St James purely because they come from Catholic families.
Also, lots of dumbass parents who think it's more elite to send their kids to a out of state school like mizzou, colorado, iowa or SLU than to go to a ku/k-state.