Theyre the same thing. Its just that adaptive optics "flatten out" the distorted wavelength of light entering in the telescope. Thus, by allowing light to pass through the telescope undistorted, it drastically improves image resolution. The adjustable mirror can be a single thin piece or it can consist of individual segments. They are adjusted by piezoelectric actuators.
Active optics involves correcting the distortion of the large mirrors due to temperature, rather than compensating for the turbulence in the atmosphere. Larger movements are require and are controlled by actuators and is much slower as adaptive optics samples light up to 1000 times per second.