Hi JoshNorman,
I'm not quite sure how you got the 2 screenshots you posted. You state both are Path Tracing renders, and mention the visible environment being used in the second image (although its not plugged in), but I can't figure out how you've set it up.
I tried a few tests, and as far as I can tell, your second image just seems to be using "direct lighting", not path tracing.
First off, the 2 screenshots have completely different environment image rotations, which will lead to different shadows, so I rotated the environment image to match your second screenshot (with darker shadows)
Here are my results.
With your Path tracing settings, I obviously couldn't match the look of your second image:
I can get a closer match to your second image by setting the kernel to direct lighting:
I can also get something similar with path tracing if I set diffuse depth to 2, although the shadows aren't as dark. There isn't much point using path tracing this way though: