Dnk,
Where did you hear of one of them?.
My experience tells me either Star-Delta (where the motor starts in Star and runs in Delta).
But a Star-Star starter?.
The reason behind the Star-Delta start is the lower current upon start-up, hence a lower voltage drop on the supply lines.
And Star-Delta requires 3 contactors.
Having said that, I can find no advantage in starting a motor in Star and also running it in Star.