"Do the right thing" defined as “– follow the law, act honorably, and treat each other with respect,” just doesn't feel the same.
"Don't Be Evil" appeals to a force above the law, sometimes the law is evil (Nazi Germany).
"Do the right thing," implies there is a single right thing which is woefully out of touch with the vast array of choices we have each day that fall all along the moral spectrum. It's cliche; I have trouble not immediately writing it off.