I've had burdock root in Japanese cooking, and dandelion root brewed straight-up in infusions. Not much of a fan of either, flavor-wise - dandelion doesn't seem interesting enough to use for a commercial product, and burdock's a bit repellent to me - but maybe they redeem each other in soft drinks. If I found some I'd definitely try it.
Root beer - a Left Pond soft drink, by name - has a distinctive, robust flavor that I can't compare to anything else in my experience; it shares some subtle elements with wintergreen, but the relationship is far too distant for me to sustain the comparison. "Fruity" might apply if you stretch the meaning to within an inch of its life, but it misses the mark by a mile, because there's an up-front, warm, earthy-spicy, almost myrrh-like razzle to the flavor that defies "fruity" in the way we might usually think of it. In the US, root beer's a nostalgic taste from childhood, it's always readily available in the shops, adults still drink it - I do from time to time - and it's its own beast altogether. Which brings us to sarsaparilla:
What was called "sarsaparilla" in 19th-century US would actually have been basic root beer as I know it: sassafras root and birch oil, not genuine sarsaparilla, were the flavorants. So technically "sarsaparilla" was a misnomer at the time: actual sarsaparilla is a tropical plant that probably cost too much to be generally affordable on a large scale, but the flavors of sarsaparilla and our much more readily-gotten native sassafras root are favorably compared, hence the substitution. I'm wondering, Ben, which kind was the sarsaparilla you had; but if you ever taste Left-Pond root beer in the future, I'm sure you'd recognize the flavor.
In the US, the word "sarsaparilla" is culturally now almost exclusively confined to Westerns where it carries a quasi-mythical status because most folks nowadays don't really know for sure what it is or ever was anymore, but as a viewer you're expected to take the undefined aspect of it at face value. All you need to know is that it's supposed to be thought of as quaint and outmoded, and whatever it is, it ain't booze. This mysterious potion of a bygone era marks the teetotaler - a dissonant element in a screen milieu where putative real men drink whiskey. Otherwise in daily use we've abandoned the name, no doubt in no small part because there's no actual sarsaparilla involved. Truth in advertising, I suppose. In the older Westerns, especially cartoons, your dusty, weatherbeaten, gap-toothed muleskinner caricature might pronounce it "sasspa-rilly", blurted as an incredulity when someone else orders it: A grown man doesn't order "sarsaparilla" in the Wild West, not without suspicion of being not quite fully formed, and of being possibly a milquetoast out of his element, whereupon a surprise contrary to expectations is often anticipated by the viewer - much like Columbo in his way, now that I think of it. The old black-and-white film cowboy hero and children's icon of wholesomeness, Hopalong Cassidy, drank nothing but the canonic so-called "sarsaparilla". But then he wore all black, too, both anomalies among the genre's usual iconography for the buckaroo, and for the good guy, respectively. In my cultural milieu, "root beer" would never fly in a Western, for the truer name cannot be divested of its association with carefree happy children, which would obviously deflate the heroic cowboy narrative beyond any help; "sarsaparilla" it must be, then, if we are to take the story line's Wild West credentials seriously.
In Westerns, "sarsaparilla" is the root beer that dare not speak its name.
Of course, these days your average root beer is more likely to be composed of artificial flavoring, but the flavor's pretty consistent in any case.
"If you take music out of this world, you will have nothing but a ball of fire." - Balochi musician
It's a bit unusual, but not in a way that tends to invite strong dislike. It's got a hint of liquorice to it, and I wonder if it might be a bit like sarsaparilla. I've never tasted that, but it could be in the same ball park. Of course, the pop version of dandelion and burdock doesn't taste anything like the real stuff, which is an alcoholic drink I've only ever had once. 
