In general, to notice means to realize or get something into our focus of attention (a sensory or intellectual realization or reckoning of something), and thus mostly associated to a passive behavior of receiving or observing a stimulus. It doesn't involve more than the reception/observation and the realization that comes from it.
In contrast, to heed means to take notice, to adjust or align your behavior or course of action bearing in mind some warning, advice, guidance, etc., therefore implying a more active behavior. Someone may notice something, but not heed it. As in these examples,
Anthrax Scare: CDC Lab Didn't Heed Its Own Lessons - NBC News
Survival Rises When Hospitals Heed All Heart Attack Guidelines - HealthDay.com
Eclipse watchers who didn’t heed safety advice now feeling effects - 7News
State Dept. didn’t heed security warnings - MSNBC