Awesome. I had no idea. I feel smarter. 
To me, the classic (60s) frosted mug A&W (or the gallon jug equivalent) was most notable for the foaminess (?). It was a rich, smooth, feel to it that helped the flavor to stick to the tongue. The canned stuff has never been like that.
A lot places nowadays throw foaming agents into things but all these are quite noticeable in terms of having a soapy taste. Whatever A&W was doing, was the right way to get things foamy.
(I miss Mama Burgers.)
A lot of the richness in A&W is due to the high vanilla content. I’ve made award-winning rootbeer by combining cream soda extract with root beer extract in a 1:3 ratio. People commented that it “tasted like the stuff they get at A&W.” High praise.
Late to the party, but Koxinga and silenus nailed it. Fresh from the tap A&W is a real treat.
The A&W I recall from my teen years when it was worth a trip to the mall for a root beer float - had more of a caramel flavor than it has now. It was wintergreen + warm buttery caramel.
Now I agree that wintergreen + citrus oil describes it. Sassafrass too, maybe - that sort of perfumey bit . . . but the warmth and roasted sugar bits seem to be gone.
We still have a drive-in A&W in Vermont (in Middlebury, about 50 miles from where I live), and it still has roller-skating carhops, the window trays, and the frosted mugs. We go there whenever we are passing through the area.
Huh? Do they no longer make Mama Burgers where you live? It’s still on the Canadian menus, along with all the other ‘family-themed’ burgers. Seems part of the charm of eating there; would be weird without it.
Y’know, I just now realized that I’d responded to later comments in this thread, but never really thought about the question posed in the OP. Hmm.
Here’s my take. A&W is your “default” root beer taste, and probably is for most Americans. Hires beat them to market, homebrews beat both, but they were the market leader for ages (outside of a few years where Barq’s temporarily led due to strong marketing). Root beer started as a sweetened homemade sassafras & herb/spice brew (often containing sarsaparilla), with heavy use of wintergreen and vanilla, but over the years as the flavor became more commercialized, it settled on wintergreen, vanilla and spice components to add complexity. Spices can include cinnamon, sassafras, sarsaparilla, anise, etc. A&W seems particularly low on anise and such, mostly cinnamon/cassia IMO.
Anyway, A&W is also one of the more simple root beer flavor blends; heavy on vanilla and wintergreen, a little bit of spice. Not that “simple” is a bad thing, of course. A&W is a really mellow, sweet soda with just a hint of spice.
If A&W is your “default” taste for root beer, then other brands of root beer must necessarily be compared to this default in order to see how well they work. This will be due to mouth feel and flavor profiles mostly, while affected a little bit more by things like storage and shipping.
If a brand pushes their flavor profile too far in any one direction, they end up with cream soda (more vanilla), sarsaparilla/ginger beer (more spice), or birch beer (more wintergreen). So they have to stay in a narrow range, tweaking the spice blend and balancing the vanilla and wintergreen to differentiate themselves from A&W yet still be recognized as root beer. However, since A&W is the default, any tweak will seem “off.” Barq’s “bite” is a stronger anise flavor (and caffeine), and most other brands seem to lower the vanilla content and play with the spices a lot (ginger, birch, cardamom, black pepper, and even chili pepper), which will definitely taste different than what you’re expecting. Other brands try messing with the mouth feel by reducing or increasing carbonation, again affecting the overall taste.
Overall: there is a narrow range of flavors acceptable in root beer, and A&W is the default “norm.” To differentiate a competing brand means to alter the flavor profile in a way that it stays within the narrow range, yet differs from A&W… however, this also means differing from the norm, which will often negatively affect the experience. A similar thing happened in the cola world with the Pepsi/Coke rivalry causing the New Coke debacle-- original Coca-Cola was the “norm;” despite the sweeter Pepsi/New Coke flavor profile being more popular in taste tests, by deviating significantly from the accepted norm of what “cola” should taste like, New Coke bombed and Pepsi has remained in second place behind the standard-bearer.
I’ve drank a lot of A&W. One of the blander root beers. I prefer IBC when I can find it. Both are good.
or GA* Why? Because someone’s being PC.
Not around here. We just have the little "half-A&W"s (shared with a KFC), or even worse a food court franchise. Checking online menus locally, there’s one that has a papa burger but not the mama or baby.
In my travels when in the western US, the few times I could get to an A&W in recent years, no “family” burgers at all.
Why does A&W root beer taste like that?
Because the A&W employees spit in it 
Interestingly owned by the same company, the Dr Pepper/Snapple group. One of four of their current root beer brands… they produce A&W, Hires, IBC, and Stewarts.
(They also have four ginger ale brands in production, and two grapefruit sodas. they’re masters of niche, if nothing else.)
Speaking of Hires, is it still around? I haven’t actually seen it in stores in ages.