But, in answer to your question, honors is considered more rigorous than advanced. The typical ps sequences are cp, advanced, honors, AP.
I have to disagree with this. Many top prep schools now use the Advanced designation in lieu of AP because they don't follow College Board syllabi even though the courses prepare students for the AP tests. For example, Pomfret, a top boarding school in Connecticut, does not offer any AP-labeled courses, but they do offer Advanced courses in Bio, Chem, Physics, Environmental Science, Calculus, Statistics, US History, Psychology, Comparative Gov't, and Advanced English III and IV, all which are clearly intended to prepare students for AP exams if they choose to take them. Groton offers AP Bio and Chem, but labels their AP-equivalent Physics classes as Advanced rather than AP (although the course catalog states that students will take the exams).
At some schools the term may also be applied to courses at or above the level of an AP class, but for which there isn't an AP test. Groton also offers a course in Advanced Math Topics for which Calc II is a prerequisite. Philips Exeter offers Advanced Chemistry, Advanced Biology, and Advanced Physics, all of which explicitly claim to prep students for the AP exams, as well as Advanced Astronomical Methods. At Choate the levels offered in the Chemistry sequence include: Chemistry, Honors Chemistry, Advanced Inorganic Chemistry, Advanced Organic Chemistry, and Advanced Topics in Organic Chemistry. They also offer courses such as Advanced Astrophysics, Advanced Robotics, and Advanced Application Development.
Some prep schools do not use the "honors" designation at all, and "advanced" is applied to both AP-equivalent courses and what would be considered honors-level nonAP courses. For example, "regular" and "advanced" are the only two levels offered at Fieldston and Berkeley Carroll, two of the top prep schools in NYC. Advanced Calc I & II are AP-equivalent, while Advanced Precalculus would be honors level.
IMO, the "Advanced" button on the Common App primarily exists for students at schools like these, where the Advanced designation tends to replace AP and/or Honors labels, and may even refer to courses above AP levels (e.g. Advanced Calc IV, Advanced Organic Chemistry).