After glancing over questions tagged with both, it seems to me that:
printer relates to a printer, e.g. physical hardware device that creates images on paper, questions related to printer drivers, printer maintenance, printer consumables, etc. Usually such questions mention a specific printer.
printing relates to a process of creating such hardcopy images on paper, e.g. printer spoolers/queues (CUPS, Windows spooler, etc.), support of printing in different OSes/applications, and these questions usually do not mention a specific printer model, as they are more generic.
I think it is useful to have both tags.