Overall, it shouldn’t matter either way as far as college applications. They will see the highest level completed, and that should be Precalculus or higher. If you might have to move at some point during high school, that could be a consideration, as switching between the tracks might cause some gaps or duplication.
Do either of the schools have honors versions of their math strand? Our public HS has a compacted track where kids can take Math II, Math III, and Precalculus in 2 years. (A similar track compresses 7th grade, 8th grade, and Math 1 into the two years of junior high.)
The switch from the traditional to integrated math sequence became much more common in CA with the switch to Common Core, so they are somewhat related. However, Common Core concepts (more multi-step word problems, explaining your answer in words, etc) can be taught in either the traditional or integrated sequence. (http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf, starting around page 57)
Articles you might want to look at for the prevalence of each sequence include these:
http://edsource.org/2014/districts-split-between-choices-for-high-school-math/66169
http://edsource.org/2015/california-districts-moving-to-new-integrated-high-school-math-pathway/89288
If the private school is not only keeping the traditional sequence, but also keeping the traditional curriculum (fewer word problems, more drill), that might cause some difficulty when taking the SAT. (Though high-achieving math students, if she is one, are typically pretty adaptable to different types of challenges.)
I agree with mathyone that if she’s looking at STEM, you’ll want to see what options are available at the two schools along the lines of Robotics, Math club, Science Bowl/Science Olympiad, computer science classes, etc. In CA, the good public HS often have much better opportunities than the privates. It’s not like the Northeast; privates are often not better academically in CA.
Also, I’m not clear on why going out of state for Purdue or VaTech is the plan. My impression (CA native bias) is that they are about as good as a mid/lower-tier UC or an upper tier CSU. CalPoly SLO would be better and cheaper.