Children living in Los Altos Hills would either attend Los Altos Schools (Link to new boundary map is http://www.losaltos.k12.ca.us/schools/PDF_Files/Adopted_atte ) or Palo Alto schools (link to boundary map is http://www.pausd.org/parents/schools_sites/downloads/Attenda
The public schools in Los Altos and Palo Alto school districts are considered some of the top school districts in the state with high standardized test scores and a lot of parent involvement. Here is a link to the recent most APR test scores for Santa Clara County: http://api.cde.ca.gov/AcntRpt2008/2007Base_Co.aspx?cYear=&am
The Bullis Charter school (a public K-6 grade) which is located on the Egan Middle school campus also provides public education to students from Los Altos Hills.
Pinewood School (Private) also houses its classes for grades 7-12 in the upper campus which is located in Los Altos hills. There are also several private schools located in Los Altos (the flats) that also cater to Los Altos hills students.
Generally, Palo Alto's schools are considered to be slightly better than those in Los Altos, although both are excellent. A client of mine who is a professor at Stanford noted that the variation between student scores within the student body within a given school in either district is greater than the spread between the median between the two districts. In short, if your child is a good student, he or she will get a good education in either Palo Alto or Los Altos districts.
You will pay about a 5% premium for a property in Los Altos Hills that attends Palo Alto schools, especially Gunn High School which is the most highly ranked and is highly sought after. Additionally, the Gunn district takes in a desirable part of Los Altos Hills that is closer in on the east side of 280, where the homes have flatter more useable lots.
You actually ask a nuanced question, feel free to contact me if you would like to go into more detail.
Generally the elementary schools in LAH are highly ranked. The high school on the other hand is not as good as Gun high (palo alto). You can find homes in LAH with Gun high school. You'll have excellent schools all throughout