According to Webster's Dictionary of Synonyms:
act, behave...are comparable when used with reference to the way in which a person or thing does what is expected or responds to external influences or circumstances.
act: the most general word among those that are used with reference to the way in which a person or thing does what is expected or responds to external influences or circumstance. <how did the child act when you called him?> <he acted as if he were about to cry> <he acted frightened> <how should this powder act when mixed with water?> <this medicine acts as a poison to some persons>
behave: is widely applied to persons and their conduct with reference to a standard of what is right or proper or decorous. <one must keep one's contracts, and behave as persons of honor or breeding should behave> However, in or parallel to technical use behave often approaches act in generality. <study how steel behaves under stress> <two men may behave like a crowd...when their emotions are engaged>
Action versus Act
Action and act agree in designating something done or effected.
Action refers primarily to the process of acting; act refers to the result, the thing done. An action is usually regarded as occupying some time and involving more than one step; an act is more frequently thought of as momentary or instantaneous and as individual <the rescue of a shipwrecked crew is a heroic action; the launching of the lifeboat, a brave act> <a course of action> <the springs of action> <an act of vengeance> <caught in the act>
In the plural, action has frequently an ethical connotation and is loosely synonymous with conduct.