How is Al2Cl6 covalent and Al2O3 ionic?
You are right that the difference in electronegativity is responsible for the ionic bonds in aluminum and oxygen, and normally that would be the case with chlorine too, but $\ce{Al2Cl6}$ is a special molecule.

The structure of $\ce{Al2Cl6}$ can be viewed as two aluminum atoms covalently bonded to four chlorine atoms each. Two of the chlorine atoms bridge between the two aluminum atoms. The following is a possible representation:



A special type of bonding called 3 center 4 electron bonding is present in the structure. These bonds result from a combination of a filled p orbital and two half filled p orbitals. This causes there to be a filled bonding and a filled non-bonding orbital. The bond orders between each bridging chlorine and aluminum atom are 0.5; the structure of 3 center 4 electron bonds makes it so that the bonding orbital is delocalized over both the bonds. $\ce{Al2Cl6}$ has two of these bonds and they are responsible for the covalent nature of the molecule.