Used in English, fierce usually means animal like, aggressive, something to be feared. Angry is an emotion. I can feel angry at you which is like saying, I feel mad at you. I would not say, "I feel fierce with you." I could say, "That man is angry, he looks fierce." To describe someone who has his teeth barred or fists clenched. Fierce comes from the root, fire. May imply being out of control. Those who are angry may be able to hide their anger or control it. A fierce person is like a fire and one should stay away from a fierce being.

FIERCE http://dictionary.reference.com/browse/fierce adjective, fierc·er, fierc·est. 1. menacingly wild, savage, or hostile: fierce animals; a fierce look. 2. violent in force, intensity, etc.: fierce winds. 3. furiously eager or intense: fierce competition. 4. Informal . extremely bad or severe: a fierce cold. An animal, a cold or a storm can be fierce but anger is a human emotion. I can be angry without necessarily being fierce.

based on Cambridge advance learner's dictionary- 3rd edition: fierce can be: # physically violent and frightening # strong and powerful # showing strong feeling or energetic activity # difficult (US Informal) besides, angry: # having a strong feeling against someone who has behaved badly, making you want to shout at them or hurt them # literary describes a stormy sea or sky # If an infected area of the body is angry, it is red and painful both are adjective anyways :)