Differences between TEOS-10 and EOS-80 for salinity
From my reading of the primer and manual, I would summarize the difference as follows. Practical salinity is calculated directly from the conductivity of seawater (which is what's most often measured by buoys, floats, and other autonomous systems) whereas absolute salinity is based on the density of the seawater. Practical salinity is essentially measuring the distribution of ions in the seawater. Absolute salinity is based on the mass of all non-H2O material, whether it's ions, gases, or solids. Relating salinity to mass, rather than conductivity, makes writing conservation equations much more straightforward; a dissolution of HCO$_3^-$ into H$^+$ and CO$_3^{2-}$ changes the practical salinity but not the absolute salinity. Basing salinity on mass also makes the thermodynamics much simpler, but I'll address that in your other question.

Differences between the practical and absolute salinities crop up where there are big differences in the composition of seawater, especially in its dissolved but non-ionic components. Page 14 of the manual (Fig 2) has some maps of what this looks like: Large deviations in the lower-salinity Pacific, and large deviations at depth where density and conductivity aren't so closely related. Absolute salinity can handle measurements of seawater composition in a way that practical salinity doesn't.