For just walls(no shower/tub surround) in a well ventilated bath, green board is better than regular sheetrock.

The characteristics of "Greenboard" are commonly misunderstood by homeowners and contractors alike.

It has the same gypsum core as regular gypsum board so it is not mold or mildew resistant. The facing is a thicker paper with a wax coating but it is not water-proof.

Since there is now mold and mildew resistant wall board available and cement (concrete) backer board is required by code behind tile in wet areas (i.e. showers) I can think of no use for Greenboard and am surprised it is still made.

What you need to find out is if the contractor is specifying Greenboard and calling it mold-resistant by mistake or is specifying mold-resistant gypsum board and calling it Greenboard by mistake.

For gypsum board to be mold-resistant it must have a facing that mold cannot feed upon. The facing of Greenboard provides some additional resistance to water penetration but so does paint on regular board.

The original intent of Greenboard was to limit water penetration behind tile in showers but because it eventually fails in that situation, there is no longer a practical use for it.

Types of mold-resistant drywall:
1) no facing (USG Fiberock Aqua-Tough, etc.)
2) fiberglass mat facing (Georgia-Pacific DensArmor, etc.)
3) treated paper facing - (Georgia-Pacific ToughRock, Certainteed M2Tech, etc.)

Another approach to mold reduction in a bathroom is to limit the sources of moisture. If a bathroom is above grade an exhaust fan is usually all that is needed. If on a slab-on-grade or in a very humid climate, mold-resistant gypsum board might be needed.

Besides - the cost differential between greenboard and plain old dry wall isn't that much. I doubt the difference for a bath room would more than $100.

If you are going to paint the drywall, the vapor permeable wax coating on the Greenboard would be redundant. If you are going to cover Greenboard with tile in a non-shower area, it would also be redundant.

Times have changed. When trying to save money in construction redundant materials is the first thing to eliminate.

I would agree with Renovator8! Save the money on the green board and put the money you saved into a quality kitchen and bath paint. Once the wallboard is sealed with the paint you are good!