I have the option of graduating from my university (U.S.) a semester early, and if I do so, I would receive the lowest honors designation of cum laude. However, if I do not graduate early, I will be able to receive the highest honors designation of summa cum laude. How important is this designation for graduate school admissions, and is there a major difference between these two? My GPA would remain unchanged, the only difference is completion of other honors program requirements.
The honor designations carry essentially no weight. The additional honor program requirements, however, could carry substantial weight. If for example, one of the extra requirements was an independent research project, this would be very valuable for graduate school admissions. If the requirement was the completion of a university breadth requirement (e.g., basket weaving), then it has no extra value for admissions.
In mathematics, in the U.S., the honors degree per se doesn't matter much, but the fuller coursework (which does mostly reflect what people have learned...) is a big plus. That is, graduating early, if it means "stopping studying mathematics", is not at all a plus, but represents a misunderstanding of the enterprise. Of course, graduating early but/and continuing to study, perhaps even more energetically/aggressively, is positive... if one does not mistakenly disengage from faculty who can be good letter-writers.