Good question!
Aerosol Optical Depth (AOD) measures the light extinction produced by aerosols in the atmosphere (either by aerosols scattering or absorbing light). AOD tends to be on a scale of about 0-1 where 0 indicates there are no aerosols present (clear sky) and 1 indicates there are dense aerosols present (it may be difficult to see the sun). The aerosols impacting the AOD could be pollution from factories, smoke, dust, sea salt, and other sources.
Aerosol Index (AI) indicates the presence of UV-absorbing or UV-scattering aerosols. Positive AI values generally indicate the presence of UV-absorbing or partly-absorbing aerosols. Examples of these aerosols include black carbon, dust, or smoke. Very small or negative AI values generally indicate clouds or the presence of aerosols that are UV-scattering. Some examples of aerosols that will cause a negative AI value include sea salt or sulfate.
AI can give you more information about the type of aerosol present whereas AOD gives you more information about the density of aerosols present.