As far as most CD players are concerned; not much.  Accurately, though, a “replicated” CD is one that’s been pressed by a plant – it’s a thin layer of aluminum embedded in plastic, manufactured to have a reasonably long lifespan.  A duplicated CD is essentially a CD-R, a burned copy of your master disc.  CD-Rs are thought to have a shorter lifespan than a full replicated CD, but are also significantly cheaper to produce and are able to be produced in smaller quantities.  The shortest run for a replicated CD is usually 300, and it’s not generally financially advantageous until quantities of 500-1000 or above, whereas duplicated CDs can be done in much shorter lots.

Is there an advantage to one over the other?  It’s really a personal and financial choice.  It can be argued that a replicated CD looks more professional than a duplicated CD, and for large runs is a cheaper option.  For short runs, a replicated CD can be comparatively expensive to produce, and if you don’t think you can move 1000 units of that CD, you’ll be stuck with a closet full of CDs you paid a significant amount for.