Action is the act of doing something.
Behaviour is more about the way in which you act.
For example, when two chemicals are mixed, an action occurs (a reaction, in fact). The way in which those two chemicals (or the compound they create) act may be described as a "behaviour".
In human behaviour, a collection of different actions, possibly recurring actions, may collectively be called a behaviour. A child described as "behaving badly" at school may have a history of kicking, punching, screaming etc - all individual actions that make up a behaviour or behavioural pattern.
The fact they are not always interchangeable is evidence of them having separate meanings. For example:
To end poverty would require action.
You couldn't directly substitute this with "behaviour", although you might say...
To end poverty would require changes in behaviour.
...which might imply a series of new actions, collectively described as a behaviour.