Performance wise, cold air makes more power because you can shove more air and subsequently more fuel in a cylinder, the key here is the more fuel part. In regular part throttle driving, you don't need this improvement in air temp and increased DA so its a detriment but still occurring due to the colder denser air. At the track where you are at WOT looking for every once of power, its an advantage.
Colder denser air will always require more fuel and that same colder denser air in part throttle driving still has tuning impacts from the IAT and the MAP on the fueling table, cold air, your AFR will go up and if unchecked, past stoich 14.7:1 and lean the engine out so fuel is added to keep the engine in a safer 12-3 range. That added fuel is why the mileage drops in colder air.
ETA, FYI, that's how the handheld tuners work, they lean the engine out and increase timing, that's the reason they need you to always run 91-93 octane fuel. Leaning an engine out can increase power because OEMs use extra fuel to cool down the cylinder to protect the piston, rings and the intake valve. That extra fuel cooling a cylinder reduces power, the hotter the combustion process, the bigger the explosion, which is more power. A leaner engine can do this but theres a point at which you can get too lean and burn a piston (which is really melt, crack or break a ring land).
