I can't speak to the social history behind it, but on a legal level, a ton changed. The Espionage Act of 1917, expanded by the Sedition Act of 1918, was the primary tool used against anti-war activists, including Eugene Debs (who you reference above). On a legal level, the major development was moving from the "clear and present danger" test to the more nuanced and less easily-abused Brandenburg test. As lawyers, we care less about what specific acts are legal/illegal, and more about what tests are used to declare an act il/legal. So I'll deal mostly with the tests, and try to show why the specific acts are less important.

Schenck and the Clear and Present Danger Test
The 1917 Act was dealt with directly by SCOTUS in Schenck v. United States and Abrams v. United States, with Schenck being the standard for judging claimed First Amendment violations for years afterwards.

Schenck dealt with a man who was arrested for distributing leaflets encouraging people to resist the draft. In Schenck, the Court formulated a test called "clear and present danger" to determine if the speech was eligible to be criminalized.

The question in every case is whether the words used are used in such circumstances and are of such a nature as to create a clear and present danger that they will bring about the substantive evils that Congress has a right to prevent. It is a question of proximity and degree. When a nation is at war, many things that might be said in time of peace are such a hindrance to its effort that their utterance will not be endured so long as men fight, and that no Court could regard them as protected by any constitutional right.

That bolded portion is the rule. Now, that's pretty broad, as we see in the Court's later discussion of the case at hand.

The statute of 1917, in § 4, punishes conspiracies to obstruct, as well as actual obstruction. If the act (speaking, or circulating a paper), its tendency, and the intent with which it is done are the same, we perceive no ground for saying that success alone warrants making the act a crime.

So basically, the statement here is that encouraging people to resist the draft is the same as conspiring with those people to obstruct the draft, which is punishable by Congress. Note that they don't do much in the way of assessing what a "clear and present" threat to obstruct the draft would look like; they just assume that this could have resulted in obstruction, therefor it's capable of criminalization. That "could have" leaves a lot of room for courts to credit the government's idea of what constitutes a possible outcome.

Now, the Espionage Act is still in force today (it's what Snowden would likely be charged with violating, for instance). But the 1918 expansions dealt directly with disparaging the WWI war effort, so they were defunct after the end of the war. Still, sedition prosecutions continued at the state and federal level alike, and were examined with the "clear and present danger" test of Schenck. See, e.g., Whitney v. California (1927) and Dennis v. United States (1951).

There was an argument for broader protection during this period, typified by Justice Holmes's concurrences and dissents in Abrams and Whitney. Basically he wanted to put a bigger wall around "political speech" as being specially protected. See his dissent in Abrams v. United States, where he first discusses this broader political protection and the "marketplace for ideas" as being central to the Constitutional design of the First Amendment. How he defined "political speech" is its own discussion unto itself, as evident by the fact that Holmes wrote the Schenck decision that ended up criminalizing acts that Schenck surely saw as deeply political. But that's a whole 'nother thing.

Anyway. Fast forward. The states and federal government used the "clear and present danger" test quite liberally throughout the post-WWI era to prosecute leftists and radicals, and was generally upheld. Free speech advocates grew to distrust the term, and see it as inviting broad potential scenarios in which some "danger" might exist.

Brandenburg and the New Test
In 1969, the Court decided the case of Brandenburg v. Ohio, dealing with speech in a totally different context from left-wing politics. Brandenburg was a local KKK leader, who encouraged his followers at a rally to enact "revengeance" on racial minorities and their perceived sympathizers, and called directly for violence. Now, given the broad standards of "clear and present danger," it appeared that this could be criminalized. After all, is this not the first step towards a conspiracy to commit assault, in the same way that Schenck's flyers were an attempt to form a conspiracy to obstruct the draft?

The Court created a new test, however. It found that Brandenburg's speech could not be criminalized, because it did not pose an threat of likely, imminent unlawful action. Well that doesn't sound too different from the "clear and present danger" test, but let's break it down a bit further. The test as applied by the courts has 3 prongs:

1. The speech has to be aimed to cause unlawful action;
2. The context has to be such that the unlawful action is likely to occur;
3. The likely unlawful action must be immediate or "imminent."

Note that this carries forward the "unlawful" requirement, so both Schenck and Brandenburg are rooted in the threat of something else happening that Congress/the states could ban. This is the point where that thing I said about the tests being more important than the specific instances comes into play. Specifically, Brandenburg dealt with the constitutionality of an Ohio statute which criminalized

advocat[ing] . . . the duty, necessity, or propriety of crime, sabotage, violence, or unlawful methods of terrorism as a means of accomplishing industrial or political reform

and

voluntarily assembl[ing] with any society, group or assemblage of persons formed to teach or advocate the doctrines of criminal syndicalism

So in neither case did the statute in question criminalize behavior that was "imminent," and there was nothing in the record below to reflect an imminent danger. Thus, Brandenburg could not be convicted.

Now, it may well be that there was a group of minorities on whom Brandenburg's audience could have imminently enacted its "revengeance," and in such a case, a speech exhorting them to do so could be criminalized. But that would be something other than merely advocating the necessity of crime to accomplish political reform. It'd be inciting a particular crime at a particular time against particular targets. So the Ohio statute was too broad, and as applied to the actual facts in Brandenburg, it could not meet the new imminence prong.

Brandenburg has at least nominally been the test ever since, with many a statute running afoul of the imminence prong in particular. There's been some evidence that it's weakening (particularly in the context of the PATRIOT Act) but that's well within the 20-year rule, so I'll refrain from extensive comment on that note, aside from to say that this area is far from static even now.

Another Vietnam Speech Case: US v. O'Brien
This is more of an aside, and another way of showing that the test is more important than the outcome a lot of times. In O'Brien, a man was convicted of obstructing the draft by burning his draft card. The Court upheld his conviction. Now, again, that looks bad for free speech, but what the Court said there was that burning the draft card was speech, and likely speech that would be protected by the First Amendment. Compare to poor Mr. Schenck, who was convicted for encouraging others to resist the draft, which maybe sorta could've lead to an act like Mr. O'Brien's. So for one, this case establishes that non-verbal, non-written actions can have protected speech elements. Further, O'Brien established a distinction between statutes that specifically target particular types of speech, and statutes that are "content neutral" -- that impinge on speech only incidentally, as part of serving a significant government interest. A requirement of content neutrality made it at least somewhat harder to target specific political groups, including anti-war groups. In fact, under O'Brien, it's quite likely the 1918 Sedition Act would be unconstitutional for a lack of content neutrality and narrow tailoring.

Well first off, you have to look at the difference between WW1 and WW2, and Vietnam. Both WW1 and 2 were large conventional conflicts that required massive mobilization, while Vietnam was a counterinsurgency war.

Then, you have to look at the massive social and political change from WW2 to Vietnam. When FDR was preparing America for WW2, he learned from Wilson's mistakes of the past and refused to institute repressive laws against American citizens.(With the exception of Japanese Americans) He instead used massive propaganda to convince Americans to support the war. Which basically continued on into the Cold War with Truman to mobilize Americans to hate Communism.

Following this, is the start of a new generation of Americans, the Baby Boomers. They were the "TV Generation" that watched a lot of TV and were disillusioned with their parents. So, when they started going to college beginning in 1964, they saw the Civil Rights Movement and many became radical and started actively participating in it. The Baby Boomers were already increasingly discontent with authority.

You combine all of these factors together with the mass televising of the Vietnam War and you get many outraged students at first, then as years drag on you have many non-radical Americans seeing the war as a mistake. This political climate made repression of anti-war speech political suicide.

Feel free to correct if I'm wrong here, this is a pretty general answer.

we have very popular, very public anti-war protests.

I'm almost certainly not the best person to answer this, but it's really, really important to realize that the anti-war movement of the 1960s was by no means mainstream and was, in fact, violently suppressed on a few occasions... most dramatically at the Democratic Convention of 1968, also called The Battle of Chicago.

Haynes Johnson, who'd been covering the convention for NBC's Today show, recalled:

The 1968 Chicago convention became a lacerating event, a distillation of a year of heartbreak, assassinations, riots and a breakdown in law and order that made it seem as if the country were coming apart. In its psychic impact, and its long-term political consequences, it eclipsed any other such convention in American history, destroying faith in politicians, in the political system, in the country and in its institutions. No one who was there, or who watched it on television, could escape the memory of what took place before their eyes.

Include me in that group, for I was an eyewitness to those scenes: inside the convention hall, with daily shouting matches between red-faced delegates and party leaders often lasting until 3 o'clock in the morning; outside in the violence that descended after Chicago police officers took off their badges and waded into the chanting crowds of protesters to club them to the ground. I can still recall the choking feeling from the tear gas hurled by police amid throngs of protesters gathering in parks and hotel lobbies.

...and...

Early Friday morning, a few hours after the convention ended, I took the elevator to the lobby of the Conrad Hilton Hotel, where I had been staying, to head for the studio. As the elevator doors opened, I saw huddled before me a group of young McCarthy volunteers. They had been bludgeoned by Chicago police, and sat there with their arms around each other and their backs against the wall, bloody and sobbing, consoling one another. I don't know what I said on the "Today" show that morning. I do remember that I was filled with a furious rage. Just thinking of it now makes me angry all over again.

He fared better than Dan Rather, there for CBS, who was slugged in the stomach and knocked down on the air for trying to interview a delegate who'd been ejected from the convention. Like journalists do.

Something like 700 protestors were arrested, and a famous Chicago Tribune headline read "2,000 Flee to Park in Tear Gas Attacks ".

There was also the shooting of unarmed protestors at Kent State in 1970 - four dead, nine wounded.

I'm not sure how "tolerated" this speech really was, in other words... and I'd really like to know if the previous legal justifications for suppressing speech were brought to bear in these cases specifically, or if the 1960s had more of a shoot first-find justification later mindset.