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

Is there any difference between the phrases "thrown under a bus" and "thrown to the wolves"? As far as I can tell they mean basically the same thing, but the "bus" phrases came into existence after the "wolves" phrase was already fully established, so maybe there's some connotations I'm missing.

They actually convey different meanings and are used in different contexts:

Throw under the bus:

• The clichéd expression throw under the bus means, roughly, (1) to betray, (2) to callously dispose of, or (3) to pass blame onto another for selfish reasons. It has been ubiquitous in the U.S. media for several years. While the expression might work in rare circumstances, it reeks of hyperbole and introduces violent imagery where it usually isn’t called for.

• In our search for examples in the news, about half the instances of under the bus dealt with actual vehicular violence, which to us confirms that the expression is not just overextreme but insensitive. Granted, there are many common expressions that evoke violence (including the synonymous stab in the back), but this one is worse because it’s so ubiquitous.

• If betray isn’t a good enough replacement for throw under the bus, consider double-cross, dupe, put one over, bamboozle, hang out to dry, or sell out. Some of these are themselves clichés, but at least they’re not widely overused at the moment.

(The Grammarist)

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 ]

(Dictionary.com)

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.

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.