I love this study because it 1) shows a difference in these two points even though they are relatively close together. And 2) that it corroborates the modern and ancient teachings.  Heart 7 is used for “spirit” specifically, and PC6 and PC7 have been studied extensively for autonomic phenomenon such as lowering High Blood Pressure and nausea.   Please click on link to see the abstract, I’ve excerpted and left the conclusion.
Zhongguo Zhen Jiu. 2012 Apr;32(4):325-8.

[Study on specificity of acupuncture effect of Shenmen (HT 7) and Daling (PC 7)].
Abstract
OBJECTIVE:

To observe the effect of acupunture at Shenmen (HT 7) and Daling (PC 7) on different cerebral functional regions by Functional Magnetic Resonance Imaging (MRI), and discuss the relative specificity of effect of these two acupoints…

CONCLUSION:
The activated cerebral functional regions of acupuncture stimulation at Shenmen (HT 7) and Daling (PC 7) are not exactly the same, which indicates that the acupuncture effects of the two acupoints are specific. With the same activated areas of language and cognitive function, the Shenmen (HT 7) specializes in emotion control while the Daling (PC 7) could active the autonomic nerve function area.