The conditioner is the device, conditioning is the state of the air as a result of a device. Ex. A room with a broken air-conditioner could be said to have an air-conditioner but no air conditioning; meanwhile an individual apt. with some with off central air-conditioning (for the whole building/floor), wouldn't have an air-conditioner (in the apt. itself) per se. Pretty abstract and in most cases we'd talk about air-conditioning unless you're talking specifically about getting your conditioner fixed or some other context where the device becomes relevant.