What’s the difference between Plasterboard and Dry-lining board and when you should use one or the other.

There is no difference. Dry lining is plasterboard that uses paper and white bonding to mask the joints. Once the nail or screw holes are filled the job is finished for paint or papering. Otherwise the plasterboard is skimmed. Plasterboard is UK based terminology whereas Dry-lining is more often used in America.

I would guess that the terminology can become confused where drylining board could be refered to as taper edge plasterboard, which as the name suggests has a slight taper at each long edge - much easier to tape and fill invisibly than square edge board plasterboard.