What are the pros and cons of adding a vapor barrier behind shower backer board?
You should NOT place a vapor barrier BEHIND the backer board.

Where, pray tell, would any such collected vapor/water go? 

Answer:There's nowhere proper for such moisture to egress.

The current best practice is to place waterproofing OVER the backetboard and just UNDER the tile.  Thin sheet membranes like Kerdi or NobleFlex are examples
. Paint on membranes can also be used, such as RedGuard or AquaDefense.

This a is a recent job I completed illustrating Kerdi over Hardibacker.