I purchased some purple drywall (mold and moisture resistant)for installing in a finished basement bedroom. One wall is an exterior wall. Do I need to install a vapor barrier between the drywall and the studs?
Ask yourself this: "If I install a vapor barrier, where is the water that it stops going to collect?"
If you put a polyethylene vapor barrier behind a drywall-covered stud framed wall, then the answer is that water that condenses on the poly will fall onto the wood sill plate, growing mold and eventually rotting it out.
Dedicated vapor barriers have no place in a basement wall. If the basement wall is insulated on the outside, then the vapor barrier is superfluous; otherwise, is it dangerous, as it will trap moisture against a moisture-sensitive element, as I indicated.
Here's lots of supporting evidence, since this perspective appears to be controversial. From http://www.greenbuildingadvisor.com/blogs/dept/musings/how-insulate-basement-wall
Should I include a polyethylene vapor barrier?
No. Basement wall systems should never include any polyethylene. You don’t want poly between the concrete and the insulation; nor do you want poly between gypsum drywall and the insulation. You don’t want poly anywhere. Paul Ellringer, an energy and mold consultant in Saint Paul, Minnesota, has a collection of slides showing moldy basement insulation. In most cases, these basement walls were insulated with fiberglass batts, and included two layers of polyethylene — one on each side of the studs. Ellringer calls this a “diaper wall,” and reports that most of them are a mess. “Fibrous insulation and poly are inherently problematic, and should not be used in below-grade walls,” says Ellringer. “Sometimes when you open it up, the fiberglass is soaking wet. If the house is two to four years old, the studs are often beginning to rot.”
No interior vapor barriers should be installed in order to permit inward drying.
Unfortunately, locating insulation layers on the interior often conflicts with the traditional approach of foundation water control – namely inward drying. Constructing frame walls, insulating the resulting cavity and covering with an interior plastic vapor barrier is common (Photograph 2) and often leads to odor, mold, decay and corrosion problems (Fugler, 2002; Ellringer, 2002). Also common, and prone to similar problems, is the use of “blanket insulation” often derisively referred to as “the diaper” for the odor problems associated with the approach (Photograph 3).
If you have not yet constructed your stud wall against the existing basement wall, please read those links to find out how to do it safely and properly.
Yes, you need a vapor barrier to prevent moisture from inside the home from traveling through the wall and condensing inside the insulation. See also what is the purpose of a vapor barrier.