I was able to reproduce this effect, although I had to end up using the code:
I got the following results in my table, with both fields being identical like you stated:
Reading into the help guide, it looks like it could be an issue with the fact that the "Calculate Geometry" tool lets you utilize the coordinate system that your data is currently in, whereas the "Field Calculator" tool utilizes geodesic algorithms to calculate the area and apparently ignore your Geographic Coordinate System. See the Note and Caution below:
For this reason, it would appear that ESRI is telling us that the "Calculate Geometry" tool would be more precise for this operation because it utilizes your specific coordinate system and not an algorithm that has been created for global use.
After some digging, I really don't see a way around this.