In most cases, HSS is activated only in direct flash, because the output is greatly limited.
The advantage is that it allows you use a wider aperture, hence faster shutter speed in bright daylight. Wider aperture produces shallower depth of field, enabling your subject to stand out from a blurred background.
With slower syn speed, you are forced to use a smaller aperture, which results in a greater depth of field.
High speed sync is not a necessity per say, but it's a great help when you need to shoot with flash at high shutter speeds.
This is often used in situations such as birding as the BG is often the bright sky which would give us a high shutter speed even if we stop down the aperture to f8. In such a case, if the shutter speed is forced down to the standard 1/250 maximum for most cameras, it may cause over-exposure in the bright sky BG.
With this feature, you can shoot at the higher shutter speeds and retain the correct exposure.