QuickDraw is a collection of routines built into the Macintosh ROMs that the Macintosh uses to define text and graphic objects. PostScript is a page description language that is used by the Apple LaserWriter and other output devices. QuickDraw is native to the Macintosh operating system while Postscript is not. In addition, some programs are Postscript-based, while others are QuickDraw- based. A Postscript-based program defines each object (such as a line or box) within a data file using Postscript commands. When printing to a Postscript-based printer, the program can send these Postscript commands (instead of pixels) directly to the printer (thus printing at the maximum resolution of the printer). However, when printing to a non-Postscript printer, such as an ImageWriter II, the program will print at screen resolution (or at least, not at maximum printer resolution).
Conversely, a QuickDraw-based program defines the objects in its data files using the native QuickDraw routines. When printing from a QuickDraw-based drawing program, such as MacDraw II, printing will occur at the resolution set by the print driver, regardless of whether or not the printer is Postscript-based. This is normally the maximum resolution of the printer. When printing to a Postscript-based printer, the LaserWriter print driver converts the QuickDraw routines to Postscript instructions, then sends these instructions to the printer for imaging.
Contrary to popular belief, there is no such thing as a QuickDraw printer. (There aren't any printers that have QuickDraw in their ROMS). In reality, QuickDraw printers function on a Mac because their print drivers convert the QuickDraw routines to bitmapped images.