Cable is not just the metal wire. It's also insulation, installation, cable channels, extra space at tight bends where cable flexibility is insufficient, and maintenance of said cable. Aluminum corrodes in a lot of conditions and happens to be one of the most fatigue-susceptible metals.
For these reasons, most aircraft wiring is copper and silver-plated or nickel-plated copper. Such wiring is reliable even under imperfect conditions.
But there are cases where different considerations prevail. Sometimes it's just the need to have some minimum cable thickness, for mechanical and handling reasons. Other times it's a thick bus bar where weight makes a difference and quality is easy to ensure.
So beside copper there is some steel, copper-nickel plated aluminum, copper-nickel plated steel, and pure aluminum wiring in aircraft.
In aircraft, safety is ensured and validated with engineering and testing; risks of the kind one can take with home wiring are not accepted. The price of aluminum wire is installation and maintenance complexity instead. The ends need to be cleaned and crimped or welded, sometimes the aluminum wire is terminated with a bimetallic insert. As a result, aluminum wiring is more expensive than copper, not less. It will likely require more maintenance work in the future.
Until recently, aluminum has been mostly avoided for these reasons, and used only for thick wires and bus bars. Currently there is enough confidence in quality management and maintenance that aluminum wiring is getting more use in attempts to save some weight, the A380 in particular uses aluminum wires up to 22 AWG. Such weight savings come at the cost of more parts and more work, both in production and later in service.
Specific percentages vary widely, but generally non-copper wiring accounts for more than 10% and less than 50% of overall wire weight in a production commercial aircraft.