OK, let's put the online thing aside for a minute, and look at just school rules.

I guess one possible difference is that an honor code involves the one's responsibility towards other students and/or professors, rather than towards the institution. Where there is a large power differential (as there is between a student and a university), the party with less power should feel less bound by rules/agreements. But when the power differential is smaller (as between students, or even between students and individual professors), agreements have more moral force.