Sideslip vs Freestream Azimuth Angle
Dispensing with the math, because it makes my head hurt...

Azimuth refers to geographic direction relative to a compass reference point, so azimuth angle refers to the deviation between the body's longitudinal axis and its horizontal path parallel to the earth's surface (heading vs track along the ground).

Sideslip angle refers to the deviation, referenced along the lateral axis of the body (parallel to the wings in other words), between the body's longitudinal axis and the airflow, independently of direction relative to the earth.