Difference between "thrown under a bus" and "thrown to the wolves"?

Regarding the attribution of "Thrown to the wolves" to Aesop's fable, I would suggest that that does not capture the essence of the phrase.

Thrown to the wolves has always, to my mind, had the sense of sacrificing someone else to save oneself or to save others. i.e. pushing someone out of a lifeboat if there isn't enough room or water.

The attribution I've heard, which I'm going to pursue with a professor of Russian literature in my family, is that it comes from Russian literature where a peasant would be thrown from a troika when pursued by wolves, in order to satisfy or delay the wolves.

My understanding is:
I've seen "Thrown under the bus" used as a person being tattled upon to authority. It may be just or unjust, but they are exposed for a real or false wrongdoing. The motive may be the whistle-blower out for gain at the other's expense or to simply prevent the victim of this tactic from doing any more damage to the shared situation, frequently in a workplace or school.
I've primarily seen "Thrown to the wolves" as meaning simply exposing a person to mob mentality or being abandoned in an environment where they are helpless and likely to fall victim to whatever hazards will ultimately end their endeavors unsuccessfully.
The key difference here is that "the bus" indicates one singular source of authority being the hazard, whereas "the wolves" indicates a source of danger that is at ground-level, similar in class or social standing to the victim -- basically "the masses" or the environment/nature itself, and plural in nature.

While the choice of which to use is probably best made on a case-by-case basis, "thrown to the wolves" often implies being left with some angry or otherwise unmanageable group of people who will target the person so "thrown":

The substitute teacher was thrown to the wolves with the unruly third-grade class after only 10 minutes of prep time.

This sentence would sound wrong if "thrown under the bus" was used instead. "Thrown to the wolves" is a much more apt (and funnier) metaphor for being unprepared to deal with a group of noisy, mischievous children acting out.

By comparison, "thrown under the bus" can be used for a betrayal where no specific group is eagerly waiting to punish or harass the person:

"Danny threw me under the bus last weekend. He promised me he was going to pick me up at the airport, but he spent the day trying to get back with his ex instead and never returned my calls. I had to pay $75 for a cab and lug three heavy bags up two flights of stairs by myself."

"Danny threw me to the wolves" would sound confusing and uncolloquial in this context. Comparing spending too much on a cab and lugging heavy bags upstairs by yourself, to being ripped apart by a pack of wild animals, comes off overdramatic and purple. Being thrown under a bus, by contrast, works so well as a metaphor here because it's an avoidable, very unpleasant event that someone else unfairly forced you to deal with.

Origin and sense of 'throw to the wolves'

As user66974's answer indicates, Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013) associates "throw [someone] to the wolves" with one of Aesop's fables:

throw to the wolves Also, throw to the dogs or lions. Send to a terrible fate; sacrifice someone, especially so as to save oneself. For example, Leaving him with hostile reporters was throwing him to the wolves, or If Bob doesn't perform as they expect, they'll throw him to the lions. All three hyperbolic terms allude to the ravenous appetite of these animals, which presumably will devour the victim. The first term comes from Aesop's fable about a nurse who threatens to throw her charge to the wolves if the child does not behave. {First half of 1900s}

The fable in question does, indeed, involve a threat to surrender someone to a wolf—but the fable "The Mother and the Wolf," as presented in Aesop for Children is considerably less lurid than you might imagine from Ammer's brief allusion to it:

Early one morning a hungry Wolf was prowling around a cottage at the edge of a village, when he heard a child crying in the house. The he heard the Mother's voice say:

"Hush, child, hush! Stop crying or I will give you to the Wolf!"

Surprised but delighted at the prospect of so delicious a meal, the Wolf settled down under an open window, expecting every moment to have the child handed out to him. But though the little one continued to fret, the Wolf waited all day in vain. Then, toward nightfall, he heard the Mother's voice again as she sat down near the window to sing and rock her baby to sleep.

"There, child, there! The Wolf shall not get you. No, no! Daddy is watching and Daddy will kill him if he should come near!" Just the the father came within sight of the home, and the Wolf was barely able to save himself from the Dogs by a clever bit of running.

[Moral:] Do not believe everything you hear.

Whether Greeks of Aesop's time were accustomed to threatening to give naughty or fussy children away to "the Wolf" is not of much import if English-speaking people didn't adopt the same threat and attribute it to wise old Aesop. I have no reason to suppose that they did either thing, and therefore Ammer's interpretation of the origin of "throw [someone] to the wolves" seems rather far-fetched.

John Ayto, Oxford Dictionary of English Idioms, third edition (2009) offers what I take to be a more plausible source for the expression:

throw someone to the wolves leave someone to be roughly treated r criticized without trying to help or defend them. informal | This phrase probably arose in reference to tales about packs of wolves pursuing travellers in horse-drawn sleighs, in which one person was pushed off the sleigh to allow it to go faster, so enabling the others to make their escape. [Example:] 1958 Listener This able and agreeable doctor was thrown to the wolves by a Prime Minister who had good reason to know that his own position was desperate.

It seems to me that this scenario (where people in dire straits sacrifice one of their number to save everyone else) is far closer to the modern sense of "throw someone to the wolves" than the scenario in which being given to the wolves is a punishment for having been naughty, whiny, or otherwise difficult to share a room with. The implicit assumption of the "throw someone to the wolves" situation is that, whatever one chooses to do, the wolves are going to have their meat; it's just a matter of whether their victim will be one person or another or perhaps everyone present.

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Origin and sense of 'throw under the bus'

That implicit assumption also differentiates "throw someone to the wolves" from "throw someone under the bus": buses aren't wolves, and they don't demand a victim. Therefore, although throwing (or pushing) someone under a passing bus may be a convenient way to distract the attention of a crowd of people so that you can slip away undetected, it involves making someone a victim of the bus when there was no existential necessity for anyone to be a victim of the bus.

Chuck McCutcheon & David Mark, Dog Whistles, Walk-Backs, and Washington Handshakes: Decoding the Jargon, Slang, and Bluster of American Political Speech (2014) has this entry for "Throw under the bus":

Throw under the bus: The opposite of loyalty; an all-too-frequent occurrence in which someone ditches or trashes a friend, employee, or associate.

Examples of throwing someone under the bus in politics date back to the 1990s and are legion. The Washington Post's David Segal branded it "the cliché of the 2008 campaign" and observed it doesn't make logical sense. It ostensibly refers to a campaign bus , and by extension a politician and his or her cause. Because the person being cut loose presumably is aboard the bus, how can they be tossed beneath a vehicle in which they are riding?

It was often used in the run-up to and during the October 2013 government shutdown. Congressional Republicans decided one way to frustrate Senate Democrats—and, by extension, President Barack Obama—was to make lawmakers vote to keep their subsidies for health insurance, as provided by the Affordable Care Act (aka Obamacare). But a Republican bill to kill the subsidies also affected congressional staff. "I understand it politically, and as a talking point," one Republican staffer told Mother Jones. "But Congress literally threw staff under the bus on this. ... You're hurting staff assistants who are sorting your mail."

In this discussion, it appears that the notion of "throwing someone under the bus" means to inflict harm on an ally in order to score political points for yourself. That the bus was originally conceived of as a campaign bus is noteworthy because a political campaign (in the U.S., anyway) is all about shuttling around the country by car, bus, train, or airplane, seeding a bog-standard everyday stump speech with parochial appeals to the locals at each whistle stop or bus tour destination.

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How different are the two expressions?

I have already indicated that "thrown to the wolves" involves the sacrifice of someone else under the compulsion of existential necessity, whereas "thrown under the bus involves the sacrifice of someone else under the impulsion of perceived immediate political self-interest. But just how different are those two situations in (political) practice, anyway? It seems to me there is considerable overlap between the two.

Suppose, for example, that a U.S. presidential administration makes a bad decision and the decision blows up in its face. In response, the president or someone else in the upper echelons of the administration may decide that someone lower down in the administration needs to take the rap or the fall for the bad decision—to put distance between the rest of the administration and the person blamed for having screwed up, and to demonstrate that the administration is doing something about the original error (namely, getting to the bottom of it and blaming someone for it).

So is the fall guy in this case being "thrown to the wolves" or being "thrown under the bus"? I suppose the answer depends on how realistic you think it is to view news reporters and opposition politicians as wolves. If the comparison seems apt, then "throwing the fall guy to the wolves" is a thoroughly defensible idiom to use; if the comparison seems patently untrue, then "throwing the fall guy under the bus" is a more true-to-life figurative description of what the big shots in the administration just did.