'Act' can also be used to mean behave in a certain manner.
"He acted has if he had seen a ghost."
"He acted calm."
Note that in the last sentence, if he had performed a single act, we would say "He acted calmly." The first sentence refers to his behavior in general.
Also, when we are describing behavior that is forced or false, where the person is trying to hide something, we tend to use 'act'.
"He acted as if he didn't have a worry in the world, but he was actually terrified of what was going to happen."
You could use 'behave' in that last sentence, but 'act' is used more, to give the image of an actor in a play.