A lot of taps have centres drilled in their drive end. If you drill a hole in a workpiece on a mill or drill for a tapped hole you leave the workpiece in the drilled position, put the spring centre in the chuck or collet of the mill or drill. You then position the tap tip in the hole and bring the spring centre down to the point where the centre is in the tap's centre drilled hole. This aligns the tap with the drilled hole to ensure the thread is produced evenly without an angle to the workpiece.

Another use is when setting up a centre popped hole in a workpiece in an independent chuck on a lathe. You put the spring centre in the tailstock, trap a plain centre between the centre pop and the spring centre and put a DTI on the centre. As the chuck is rotated the eccentricity of the centre pop is followed by the centre and shows up on the DTI. Much easier than trying to set the centre pop to a tailstock mounted centre or by eye.

There are probably other uses as well that other people can suggest.

Martin