Shouting is simply speaking loudly or making a loud sound. If you saw your friend some distance away at a crowded park, you might shout her name to get her attention. A soccer coach might shout instructions to the players on the field. Shouting can be angry, joyful, or neutral.
When you scream, you make a long, loud, high-pitched sound. People usually scream in moments of fear or pain. It is often a reflex action (screaming when something frightens you) but you can also scream intentionally. A parent whose small child starts to wander out into the street might scream for him to stop. People who are trapped in a car after an accident might scream for help.