Although the terms are generally synonymous, the difference between these tools is based on the usage of the pour points data.
Pour points are points where water flows out of an area, usually the outlet or re-entrant locations from the flow accumulation. The Snap Pour Point tool from the Hydrology toolset snaps these points to the cell of highest flow accumulation within a specified distance. The Watershed tool is based on the output of the Snap Pour Point tool and only selects the pixels that contribute to a particular pour point.
The Watershed tool is advantageous when analyzing familiar areas where the locations of the pour points are known. The tool allows for control over the pour point information.
On the other hand, the Basin tool is based on the ridges throughout the raster and divides the entire raster into different basins. Although the Basin tool generates similar results to the Watershed tool, it is more useful for large study areas or unfamiliar sites because it automatically selects the pour points.