We might have to agree to disagree on this one. I find that anise provides a rich and complex black "licorice" flavor to the drink, and it is very prevalent in Barq's.
Wintergreen provides a very minty flavor-and Mug is overwhelmingly minty. I like it to Pepsodent toothpaste, Wint-o-Green Life Savers, and Winterfresh gum. It provides a cooling sensation(as minty things usually do), and I detect almost none of that in Barq's.
The only other culprit I can think of is possibly Nutmeg. Nutmeg usually provides an aromatic quality to foods it is included in, and somewhat of a "sting" in the mouth like allspice and clove. While it is possible those last two are used as an ingredient in Barq's, if they are present it is very minimal, or washed out by the HFCS...
I'm still going to say it's anise or nutmeg. I'll be brewing my own batch of Root Beer soon here, and I'll be using both wintergreen and anise, maybe even nutmeg. Even so, I want to go make some teas out of anise, wintergreen and nutmeg separately to see what kind of flavors/mouth feels/aromatic qualities each as.