What is the difference between AHRS and ADIRS?
The most important difference is that an ADIRS contains an inertial system, which gives a position estimate and can be used for navigation. This is done by aligning to a known position on the ground and then integrating all changes in orientation (measured by ring laser gyroscopes) and velocity (measured by quartz accelerometers) to update the position. As a by-product you also get the orientation (attitude and heading).
The AHRS only gives this orientation information, but no position. It therefore replaces conventional gyroscopic instruments like a directional gyro and an artificial horizon. An ADIRS can usually be switched to ATT mode (instead of full NAV mode) in flight and then function as an AHRS without navigation capabilities:

(source: 737 NG FCOMv2 (11.10.21 - Flight Management, Navigation - Controls and Indicators))
The ADIRS also performs all computations done by an Air Data Computer (ADC), which usually comes installed separately next to an AHRS for glass cockpits, and therefore replaces the conventional pitot-static instruments.