Hi Andrew - apologies for the long delay on this, and for the confusion. We need to make a change in the software as well as documentation to address this. Both measurements should be correct, but we need to clarify what is being measured.
As you've observed, the VOLUME tool also reports the area of the polygon you've entered. In this case, it is reporting the the planimetric (2D) area of the polygon projected onto a flat map.
The AREA tool actually calculates the surface area inside your polygon. So, if you think about measuring a gravel stockpile, the surface area tells you how many square feet of paint would be required to cover it. If you test a flat area, the numbers should be close, but otherwise the AREA tool is always going to report a larger value.
We're working to clarify this in the documentation, and my hope is to update the software to define and report both area measurements, no matter which tool you use. I feel both have value, but we need to be explicit about what is being reported.
Cody B.