Difference between any arene and an aromatic compound?
Arene — Compound which contains a benzene ring

Aromatic Compounds — Compounds having aroma (wait, the modern definition is different, I know)

Every arene is an aromatic compound but every aromatic compound need not be an arene.

Mathematically, 

$$\text{arene} \subset \text{aromatic compounds}$$

therefore anything which contains a benzene ring has to be aromatic

Some examples of arenes:



Getting into aromatic compounds, the modern criteria for aromaticity are:


The molecule is cyclic (a ring of atoms)
The molecule is planar (all atoms in the molecule lie in the same plane)
The molecule is fully conjugated (p orbitals at every atom in the ring)
The molecule has $4n+2$ π electrons ($n=0$ or any positive integer)*


Examples (non-benzenoids): 



Now, benzenoids (compounds which contain benzene) do match with this. Therefore, benzene derivatives are aromatic compounds.