Julie, as Jason said, there are (at least) two reasons. One is when the part is difficult to clamp because of size or shape, the second is when you want to do further turning on the item after drilling. This (a current work in progress) is an example that illustrates both issues.
Item is brass, raw stock is 30mm diameter so too big for my ER collets in the lathe. Finished diameter of the disk is 22mm, with three 2.5mm partial holes round the periphery. Rod is 8mm. Whole thing is about 15mm tall. It’s not untypical of the size of things I make.

Turn the 8mm part down on the main stock. I don’t want to try drilling those partial holes, so I will ( not finished yet) transfer the part in the chuck to the mill. Drill the three holes, then back on the lathe to turn down to the finished diameter as an interrupted cut when I reach the holes.
It can then be flipped round, the 8mm part held in a collet and the spare stock parted off.

The part would be a bit of a pig to clamp anyway because of its size. But I need to keep it all concentric as possible. I’m sure with some creativity I could have done it another way. Tool post drill and indexing system for the lathe spindle come to mind, but considerably more expensive than the <£4 delivered the blanking plug cost me.
John, understood about the unscrewing issue. Sherline suggest, and make a gadget for on these small chucks, putting a tommy bar in a hole in the body (not the scroll!) and clamp that to prevent rotation. A simple unit to do so is on the round tuit list.