Most private schools don't do marching band, or they do a much smaller version than what Texas public high schools are doing. Less rehearsal time, fewer (if any) Friday night games, it's just not as big of a production. That comes with a much lower salary (which might be worth it to you).
More freedom to do what you want
No STAAR test
Much more of a family environment - especially in a school that's k-12, or one that's been established long enough that your kids' parents (potentially grandparents) went there.
Much less funding (almost always)
No contract - you can be fired a lot easier, any time of year.
At a religious school, you have to follow their interpretation of the religion or risk getting fired.
You're less likely to have a full schedule of band. You might have to teach choir, strings, or other subjects as well. In a lot of private schools, you're the only music person on campus. Sometimes it's a part-time position or shared between a couple of different schools.