Do copper and aluminium have about the same conductivity?

Not exactly. The resistivity of aluminium is  65% higher than that of copper. As a result, to conduct the same electrical current, an aluminium cable will need a cross-section that is 65% larger than that of a copper cable.

But that's not the whole story. While less conductive, aluminium is also three times lighter than copper. As a result, copper and aluminium each have their own  application areas.