What is the difference between nonlinearity and chaos, or . . . .

Not all nonlinear systems of eqs are chaotic.

 

You're asking the question the wrong way round. A more logical question would be "under what conditions are they the same?"

As it stands, the shortest answer would be "they're not the same thing". Chaos is a different phenomenon to nonlinearity, but sometimes arises from it.

 

What is sure, is that chaotic systems are non-linear.

The definition of a chaotic system is:
- exponential divergence of initially close orbits (Liapunov instability)
- the state of the system remains in a compact space

Well, you cannot satisfy both with a linear system: in the case there is exponential divergence, at least one solution will become unbounded.