My and my son’s experiences were with an Honors Program which had many Honors courses, Honors sections and ability to do more to get Honors credit for those who worked out an acceptable plan to do so. Excellent sequences in math, chemistry physics and biology available. Also likely excellent integrated liberal arts. Plus many other courses. I agree it depends on the school.
I have done online searches of several Honors Colleges/Programs. Much variation. Not impressed at all with some that segregated students into a “college” with required generalized courses plus housing but only for the first year or two. General survey courses, whether it be in a science or other field, lack the in depth study of a subject.
Honors is not always correlated with difficulty. And difficulty depends on the background knowledge as well as the ability in any given subject on the part of the student. As in the above post, #9, choosing honors over regular also depends on goals. Theory or problem based math for example.
No comparison to HS.