Why would a flat sheet of paper and a wad of paper with the same mass not fall through the air at the same rate?

The flat sheet of paper has more surface area than the crumpled ball

The flat sheet has a larger surface area, so there will be more air resistance upon it. You have two forces on your paper, gravity (down) and normal (up.) Although gravity is stronger than the normal of air, the difference between the down force and the up is smaller. For example:

If gravity is 10 newtons and the air resistance is 5, then the force is:
10N-5N= 5N 

If gravity is in a vacuum, then air resistance is 0:
10N-0N= 10N

In the first example, you get half as much force bringing your paper down, so it will fall at a slower rate. 

When the paper is crumpled, the air resistance is lower because their is less surface area for the normal to work on.

If gravity is 10 newtons and the air resistance is 2, then the force is:
10N-2N= 8N 

More force, falling faster.

Ignore this if you don't want to be confused: 
Technically, they do fall at the same rate, because all objects fall at the same speed regardless of mass. The illusion of time difference comes from friction.