What could one say the goblet squat does that the barbell squat doesn't do
Good question.
What we do know is in the changes in Grip Width or Incline Position in an Upper Body Exercise, Foot Width Stand in a Lower Body Exercise, High Bar Bell or Low Barbell Squat, etc. turn the Exercise into a completely different movement.
Research
EMG Research would provide one of the most effective method in determining the differences between a Goblet Squat vs a Barbell Squat.
A more common method that will provide some empirical data is...
The Vince Gironda Method
1) Take some time off from a movement, like the Goblet Squat or Barbell Squat.
2) After a few weeks of detraining perform only the Exercise in question.
3) Preform 10 Sets of the Exercise in question.
4) Each Set needs to be perform for Moderate High Repetitions.
4) Push all of the 10 Set to failure.
5) Once that is accomplished, go home.
6) In 24 to 48 hours, the DOMS (Delayed Onset of Muscle Soreness) will provide you with some great feed back on the muscle engaged in the movement.
It Works
In working with a woman years ago, she wanted to work her glute more. She whined that the Lunges that I had her doing weren't working her glutes; she didn't feel it. My reply, "Talk to me about in your next training session in two days".
Two days later she showed up with happily showed up with a sore a#@ wanting to do more Lunges.
Kenny Croxdale